Ole Gunnar Solskjaer’s side have made an underwhelming start to their first full season under the Norwegian’s management following a return of five points from their opening four games.
Manchester United were 4-0 winners against Chelsea FC in their season opener but the Red Devils have lost to Crystal Palace and drawn with Southampton and Wolves.
Chelsea FC registered their first Premier League win under club legend Frank Lampard in a 3-2 victory over Norwich City at Carrow Road last month.
However, the Blues boss is still waiting to secure his first victory at Stamford Bridge following successive stalemates with Leicester City and Sheffield United.
Everton have collected seven points from their opening four games to secure sixth spot in the table ahead of both Chelsea FC and Manchester United.
Former England manager Allardyce believes Everton could finish above Chelsea FC and Manchester United this season.
“I think Everton should break into the top six and if they’re going to break in, this is the season to do it,” Allardyce told talkSPORT, as quoted by Metro.
“You’ve got the difficult situation at Chelsea and I feel at Manchester United as well.
“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 think Man United might find breaking in a bit of a struggle, so will Chelsea, so that gives an opening and an opportunity for a team like Everton.”
Manchester United signed Swansea City winger Daniel James, Crystal Palace full-back Aaron Wan-Bissaka and Leicester City defender Harry Maguire in the summer.
The Red Devils haven’t won the Premier League title since 2013 when Sir Alex Ferguson was still in charge of the 20-time English champions.
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