Hi lynnfromgreenwood,
It sounds like you're getting messages on one device, but there's a delay on the other device. We're glad you've joined us in Apple Support Communities. Let's see if this helps.
If you haven't already, let's restart both your iPhone and your Mac and see if the issue remains. A restart is a great step when experiencing unexpected behavior.
If the issue remains after following those steps, let us know more specifics about how long the delay is in seconds, minutes or hours.
We hope that helps with your question and take care!