Pulis has no plans to move on, while Moyes has stayed put even when linked elsewhere - yet that big job has not come his way.
Pulis said: "The Everton fans will be absolutely delighted that David has stayed when he might have been offered another job.
"But I'm surprised that whenever a top job comes up, he is not in the frame.
"I think he has proved at a big club - and Everton is a big club - he can buy well and sell well.
"He's put his stamp on the football club."