The former Blues defender has also called on the club to seek fresh investment if they are to continue to improve each season.
"The club has some big decisions to make. Unless we get a new investor it is going to be very difficult for the manager. I think David has come to a crossroads in his career," Stubbs told the Daily Star.
"Does he stay at Everton, a club he loves and where he has a fantastic relationship with the chairman, or does he take that next step?"
Everton currently sit seventh in the Premier League table.