diff --git a/src/mcp/client/streamable_http.py b/src/mcp/client/streamable_http.py
index 57df64705..d610559f2 100644
--- a/src/mcp/client/streamable_http.py
+++ b/src/mcp/client/streamable_http.py
@@ -251,47 +251,61 @@ async def _handle_resumption_request(self, ctx: RequestContext) -> None:
                     await event_source.response.aclose()
                     break
 
+    async def _send_error_response(self, ctx: RequestContext, error: Exception) -> None:
+        """Send an error response to the client."""
+        error_data = ErrorData(code=32000, message=str(error))
+        if isinstance(ctx.session_message.message.root, JSONRPCRequest):
+            jsonrpc_error = JSONRPCError(jsonrpc="2.0", id=ctx.session_message.message.root.id, error=error_data)
+            session_message = SessionMessage(message=JSONRPCMessage(jsonrpc_error))
+            await ctx.read_stream_writer.send(session_message)
+
     async def _handle_post_request(self, ctx: RequestContext) -> None:
         """Handle a POST request with response processing."""
         headers = self._prepare_request_headers(ctx.headers)
         message = ctx.session_message.message
         is_initialization = self._is_initialization_request(message)
 
-        async with ctx.client.stream(
-            "POST",
-            self.url,
-            json=message.model_dump(by_alias=True, mode="json", exclude_none=True),
-            headers=headers,
-        ) as response:
-            if response.status_code == 202:
-                logger.debug("Received 202 Accepted")
-                return
+        try:
+            async with ctx.client.stream(
+                "POST",
+                self.url,
+                json=message.model_dump(by_alias=True, mode="json", exclude_none=True),
+                headers=headers,
+            ) as response:
+                if response.status_code == 202:
+                    logger.debug("Received 202 Accepted")
+                    return
 
-            if response.status_code == 404:
-                if isinstance(message.root, JSONRPCRequest):
-                    await self._send_session_terminated_error(
-                        ctx.read_stream_writer,
-                        message.root.id,
-                    )
-                return
+                if response.status_code == 404:
+                    if isinstance(message.root, JSONRPCRequest):
+                        await self._send_session_terminated_error(
+                            ctx.read_stream_writer,
+                            message.root.id,
+                        )
+                    return
 
-            response.raise_for_status()
+                response.raise_for_status()
+                if is_initialization:
+                    self._maybe_extract_session_id_from_response(response)
+
+                # Per https://modelcontextprotocol.io/specification/2025-06-18/basic#notifications:
+                # The server MUST NOT send a response to notifications.
+                if isinstance(message.root, JSONRPCRequest):
+                    content_type = response.headers.get(CONTENT_TYPE, "").lower()
+                    if content_type.startswith(JSON):
+                        await self._handle_json_response(response, ctx.read_stream_writer, is_initialization)
+                    elif content_type.startswith(SSE):
+                        await self._handle_sse_response(response, ctx, is_initialization)
+                    else:
+                        await self._handle_unexpected_content_type(
+                            content_type,
+                            ctx.read_stream_writer,
+                        )
+        except Exception as exc:
             if is_initialization:
-                self._maybe_extract_session_id_from_response(response)
-
-            # Per https://modelcontextprotocol.io/specification/2025-06-18/basic#notifications:
-            # The server MUST NOT send a response to notifications.
-            if isinstance(message.root, JSONRPCRequest):
-                content_type = response.headers.get(CONTENT_TYPE, "").lower()
-                if content_type.startswith(JSON):
-                    await self._handle_json_response(response, ctx.read_stream_writer, is_initialization)
-                elif content_type.startswith(SSE):
-                    await self._handle_sse_response(response, ctx, is_initialization)
-                else:
-                    await self._handle_unexpected_content_type(
-                        content_type,
-                        ctx.read_stream_writer,
-                    )
+                raise exc
+            else:
+                await self._send_error_response(ctx, exc)
 
     async def _handle_json_response(
         self,
@@ -323,6 +337,7 @@ async def _handle_sse_response(
         """Handle SSE response from the server."""
         try:
             event_source = EventSource(response)
+            finished = False
             async for sse in event_source.aiter_sse():
                 is_complete = await self._handle_sse_event(
                     sse,
@@ -333,11 +348,14 @@ async def _handle_sse_response(
                 # If the SSE event indicates completion, like returning respose/error
                 # break the loop
                 if is_complete:
+                    finished = True
                     await response.aclose()
                     break
-        except Exception as e:
-            logger.exception("Error reading SSE stream:")
-            await ctx.read_stream_writer.send(e)
+            if not finished:
+                raise Exception("SSE stream ended without completing")
+        except Exception as exc:
+            logger.exception("Error handling SSE response")
+            await self._send_error_response(ctx, exc)
 
     async def _handle_unexpected_content_type(
         self,