diff --git a/distributed/comm/tcp.py b/distributed/comm/tcp.py
index de6cee016d8..05b03f562cf 100644
--- a/distributed/comm/tcp.py
+++ b/distributed/comm/tcp.py
@@ -193,9 +193,14 @@ async def read(self, deserializers=None):
             frames_nbytes = await stream.read_bytes(fmt_size)
             (frames_nbytes,) = struct.unpack(fmt, frames_nbytes)
 
-            frames = bytearray(frames_nbytes)
-            n = await stream.read_into(frames)
-            assert n == frames_nbytes, (n, frames_nbytes)
+            frames = memoryview(bytearray(frames_nbytes))
+            # Workaround for OpenSSL 1.0.2 (can drop with OpenSSL 1.1.1)
+            MAX = 1_000
+            for i, j in sliding_window(range(0, frame_nbytes + MAX, MAX)):
+                chunk = frames[i:j]
+                chunk_nbytes = len(chunk)
+                n = await stream.read_into(chunk)
+                assert n == chunk_nbytes, (n, chunk_nbytes)
         except StreamClosedError as e:
             self.stream = None
             self._closed = True