The 62-year-old former England manager, who signed a two-and-a-half-year deal on his appointment last December, was due to hold talks with club owner Steve Parish this week over summer transfer signings.
On Sunday he spoke freely about the need to bring in the right quality players after a season in which Palace came close to relegation, surviving largely because of an impressive run which saw them win four successive games in March and April.
But he appears to have had a change of heart, according to multiple media reports.
Allardyce was immediately linked with the vacancy at his former club Sunderland, whose manager David Moyes resigned on Monday.
Some reports suggested Allardyce is not intending to look for another managerial job after a rollercoaster season which began with a 67-day stint in charge of the England national team. He eventually resigned in the wake of a newspaper sting, later joining Palace when they were in relegation trouble.
Palace's Premier League status was confirmed after the 4-0 win over Hull City earlier this month and they eventually finished 14th on 41 points.
(Reporting by Neil Robinson; editing by Martyn Herman)
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