Everton finished 12 points off the top six last season, but Allardyce believes the Goodison Park club will never get a better chance to return to European football than this term.
“I think Everton should break into the top six and if they're going to break in, this is the season to do it," he told talkSPORT.
“United have gone even younger than Chelsea have with their squad this year; they've trimmed their squad down to the smallest number of experienced players I've probably ever seen.
“I don't think Wolves will do it because they're in Europe for the first time in a long, long time and that will take a toll on the squad, as we saw with Burnley last year. It'll be difficult for them.
“But certainly Everton, with the spending power they've had over the last few years and the players they've bought… "