diff --git a/Lib/asyncore.py b/Lib/asyncore.py
--- a/Lib/asyncore.py
+++ b/Lib/asyncore.py
@@ -104,7 +104,7 @@
 
 def readwrite(obj, flags):
     try:
-        if flags & select.POLLIN:
+        if flags & (select.POLLIN | select.POLLPRI) == select.POLLIN:
             obj.handle_read_event()
         if flags & select.POLLOUT:
             obj.handle_write_event()
@@ -146,7 +146,7 @@
         except InterruptedError:
             return
 
-        for fd in r:
+        for fd in set(r).difference(e):
             obj = map.get(fd)
             if obj is None:
                 continue
diff --git a/Lib/test/test_asyncore.py b/Lib/test/test_asyncore.py
--- a/Lib/test/test_asyncore.py
+++ b/Lib/test/test_asyncore.py
@@ -676,6 +676,10 @@
         class TestClient(BaseClient):
             def handle_expt(self):
                 self.flag = True
+                self.socket.recv(4096, socket.MSG_OOB)
+
+            def handle_read(self):
+                self.recv(1)
 
         class TestHandler(BaseTestHandler):
             def __init__(self, conn):