Crystal Palace wanted Allardyce to remain in charge after he steered them clear of relegation this season.
But the Daily Mail claim the former England manager wants to leave Selhurst Park.
According to the report, Allardyce wanted assurances before committing his future to the south London club.
Today the 62-year-old held a meeting with Palace chairman Steve Parish, which ended with Allardyce announcing he wants to quit the club.
Allardyce has previously admitted his concerns over the support from Palace’s American owners.
He only joined Palace back in December after Alan Pardew was sacked.
The ex-Bolton and Sunderland manager maintained his record as never being relegated from the Premier League as Palace went on a spring charge to avoid the drop.
Neither the manager nor the club are yet to release a statement.
Express Sport reported two days ago that Allardyce was set for crunch talks with Parish.
“As a manager of six Premier League clubs so far I have never got all I wanted. But we will sit down and make sure we cover the priorities with what we have available,” he said ahead of Sunday’s final game of the season with Manchester United, which Palace lost 2-0.
“The top end of what we want to do will be the first thing to cover and then if we can’t get it all we say let’s get on with it and move on.
“What happens will decide who we go for over the summer months.”
Allardyce’s decision comes three years after Tony Pulis also declined to stay on as manager after keeping the Eagles in the top flight.
There are suggestions Allardyce could return to former club Sunderland, who were relegated from the Premier League this season.
Sunderland fans are calling for their former boss to replace David Moyes, who has quit his post.
Watford are also seeking a new manager after sacking Walter Mazzarri.
