diff --git a/httpcore/_synchronization.py b/httpcore/_synchronization.py
index 9619a3983..328150040 100644
--- a/httpcore/_synchronization.py
+++ b/httpcore/_synchronization.py
@@ -64,7 +64,7 @@ def setup(self) -> None:
         if self._backend == "trio":
             self._trio_lock = trio.Lock()
         elif self._backend == "asyncio":
-            self._anyio_lock = anyio.Lock()
+            self._anyio_lock = anyio.Lock(fast_acquire=True)
 
     async def __aenter__(self) -> "AsyncLock":
         if not self._backend:
@@ -167,7 +167,7 @@ def setup(self) -> None:
             )
         elif self._backend == "asyncio":
             self._anyio_semaphore = anyio.Semaphore(
-                initial_value=self._bound, max_value=self._bound
+                initial_value=self._bound, max_value=self._bound, fast_acquire=True
             )
 
     async def acquire(self) -> None:
diff --git a/pyproject.toml b/pyproject.toml
index 85c787402..a6727ae68 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -44,7 +44,7 @@ trio = [
     "trio>=0.22.0,<0.26.0",
 ]
 asyncio = [
-    "anyio>=4.0,<5.0",
+    "anyio>=4.5.0,<5.0",
 ]
 
 [project.urls]