diff --git a/haystack/components/generators/hugging_face_api.py b/haystack/components/generators/hugging_face_api.py
index 8fe38c013..a9d987354 100644
--- a/haystack/components/generators/hugging_face_api.py
+++ b/haystack/components/generators/hugging_face_api.py
@@ -75,7 +75,7 @@ class HuggingFaceAPIGenerator:
     ```
     """
 
-    def __init__(
+    def __init__(  # pylint: disable=too-many-positional-arguments
         self,
         api_type: Union[HFGenerationAPIType, str],
         api_params: Dict[str, str],
@@ -179,12 +179,19 @@ class HuggingFaceAPIGenerator:
         return default_from_dict(cls, data)
 
     @component.output_types(replies=List[str], meta=List[Dict[str, Any]])
-    def run(self, prompt: str, generation_kwargs: Optional[Dict[str, Any]] = None):
+    def run(
+        self,
+        prompt: str,
+        streaming_callback: Optional[Callable[[StreamingChunk], None]] = None,
+        generation_kwargs: Optional[Dict[str, Any]] = None,
+    ):
         """
         Invoke the text generation inference for the given prompt and generation parameters.
 
         :param prompt:
             A string representing the prompt.
+        :param streaming_callback:
+            A callback function that is called when a new token is received from the stream.
         :param generation_kwargs:
             Additional keyword arguments for text generation.
         :returns:
@@ -194,25 +201,27 @@ class HuggingFaceAPIGenerator:
         # update generation kwargs by merging with the default ones
         generation_kwargs = {**self.generation_kwargs, **(generation_kwargs or {})}
 
-        if self.streaming_callback:
-            return self._run_streaming(prompt, generation_kwargs)
+        # check if streaming_callback is passed
+        streaming_callback = streaming_callback or self.streaming_callback
 
-        return self._run_non_streaming(prompt, generation_kwargs)
+        stream = streaming_callback is not None
+        response = self._client.text_generation(prompt, details=True, stream=stream, **generation_kwargs)
 
-    def _run_streaming(self, prompt: str, generation_kwargs: Dict[str, Any]):
-        res_chunk: Iterable[TextGenerationStreamOutput] = self._client.text_generation(
-            prompt, details=True, stream=True, **generation_kwargs
-        )
+        output = self._get_stream_response(response, streaming_callback) if stream else self._get_response(response)  # type: ignore
+        return output
+
+    def _get_stream_response(
+        self, response: Iterable[TextGenerationStreamOutput], streaming_callback: Callable[[StreamingChunk], None]
+    ):
         chunks: List[StreamingChunk] = []
-        # pylint: disable=not-an-iterable
-        for chunk in res_chunk:
+        for chunk in response:
             token: TextGenerationOutputToken = chunk.token
             if token.special:
                 continue
             chunk_metadata = {**asdict(token), **(asdict(chunk.details) if chunk.details else {})}
             stream_chunk = StreamingChunk(token.text, chunk_metadata)
             chunks.append(stream_chunk)
-            self.streaming_callback(stream_chunk)  # type: ignore # streaming_callback is not None (verified in the run method)
+            streaming_callback(stream_chunk)  # type: ignore # streaming_callback is not None (verified in the run method)
         metadata = {
             "finish_reason": chunks[-1].meta.get("finish_reason", None),
             "model": self._client.model,
@@ -220,13 +229,12 @@ class HuggingFaceAPIGenerator:
         }
         return {"replies": ["".join([chunk.content for chunk in chunks])], "meta": [metadata]}
 
-    def _run_non_streaming(self, prompt: str, generation_kwargs: Dict[str, Any]):
-        tgr: TextGenerationOutput = self._client.text_generation(prompt, details=True, **generation_kwargs)
+    def _get_response(self, response: TextGenerationOutput):
         meta = [
             {
                 "model": self._client.model,
-                "finish_reason": tgr.details.finish_reason if tgr.details else None,
-                "usage": {"completion_tokens": len(tgr.details.tokens) if tgr.details else 0},
+                "finish_reason": response.details.finish_reason if response.details else None,
+                "usage": {"completion_tokens": len(response.details.tokens) if response.details else 0},
             }
         ]
-        return {"replies": [tgr.generated_text], "meta": meta}
+        return {"replies": [response.generated_text], "meta": meta}
diff --git a/haystack/components/generators/hugging_face_local.py b/haystack/components/generators/hugging_face_local.py
index 178548c61..0e2c6ae5f 100644
--- a/haystack/components/generators/hugging_face_local.py
+++ b/haystack/components/generators/hugging_face_local.py
@@ -54,7 +54,7 @@ class HuggingFaceLocalGenerator:
     ```
     """
 
-    def __init__(
+    def __init__(  # pylint: disable=too-many-positional-arguments
         self,
         model: str = "google/flan-t5-base",
         task: Optional[Literal["text-generation", "text2text-generation"]] = None,
@@ -204,12 +204,19 @@ class HuggingFaceLocalGenerator:
         return default_from_dict(cls, data)
 
     @component.output_types(replies=List[str])
-    def run(self, prompt: str, generation_kwargs: Optional[Dict[str, Any]] = None):
+    def run(
+        self,
+        prompt: str,
+        streaming_callback: Optional[Callable[[StreamingChunk], None]] = None,
+        generation_kwargs: Optional[Dict[str, Any]] = None,
+    ):
         """
         Run the text generation model on the given prompt.
 
         :param prompt:
             A string representing the prompt.
+        :param streaming_callback:
+            A callback function that is called when a new token is received from the stream.
         :param generation_kwargs:
             Additional keyword arguments for text generation.
 
@@ -228,7 +235,10 @@ class HuggingFaceLocalGenerator:
         # merge generation kwargs from init method with those from run method
         updated_generation_kwargs = {**self.generation_kwargs, **(generation_kwargs or {})}
 
-        if self.streaming_callback:
+        # check if streaming_callback is passed
+        streaming_callback = streaming_callback or self.streaming_callback
+
+        if streaming_callback:
             num_responses = updated_generation_kwargs.get("num_return_sequences", 1)
             if num_responses > 1:
                 msg = (
@@ -241,7 +251,7 @@ class HuggingFaceLocalGenerator:
             # streamer parameter hooks into HF streaming, HFTokenStreamingHandler is an adapter to our streaming
             updated_generation_kwargs["streamer"] = HFTokenStreamingHandler(
                 self.pipeline.tokenizer,  # type: ignore
-                self.streaming_callback,
+                streaming_callback,
                 self.stop_words,  # type: ignore
             )
 
diff --git a/releasenotes/notes/add-streaming-callback-run-param-to-hf-generators-5ebde8fad75cb49f.yaml b/releasenotes/notes/add-streaming-callback-run-param-to-hf-generators-5ebde8fad75cb49f.yaml
new file mode 100644
index 000000000..792a6c327
--- /dev/null
+++ b/releasenotes/notes/add-streaming-callback-run-param-to-hf-generators-5ebde8fad75cb49f.yaml
@@ -0,0 +1,4 @@
+---
+enhancements:
+  - |
+    Add `streaming_callback` run parameter to `HuggingFaceAPIGenerator` and `HuggingFaceLocalGenerator` to allow users to pass a callback function that will be called after each chunk of the response is generated.
diff --git a/test/components/generators/test_hugging_face_api.py b/test/components/generators/test_hugging_face_api.py
index ce6c7ecd2..0f4be2f9c 100644
--- a/test/components/generators/test_hugging_face_api.py
+++ b/test/components/generators/test_hugging_face_api.py
@@ -186,6 +186,7 @@ class TestHuggingFaceAPIGenerator:
             "details": True,
             "temperature": 0.6,
             "stop_sequences": ["stop", "words"],
+            "stream": False,
             "max_new_tokens": 512,
         }
 
@@ -208,7 +209,13 @@ class TestHuggingFaceAPIGenerator:
 
         # check kwargs passed to text_generation
         _, kwargs = mock_text_generation.call_args
-        assert kwargs == {"details": True, "max_new_tokens": 100, "stop_sequences": [], "temperature": 0.8}
+        assert kwargs == {
+            "details": True,
+            "max_new_tokens": 100,
+            "stop_sequences": [],
+            "stream": False,
+            "temperature": 0.8,
+        }
 
         # Assert that the response contains the generated replies and the right response
         assert "replies" in response