The Eagles, who have been hunting for a new boss over the last week, were close to appointing Malky Mackay, but allegations of offensive text messages between himself and ex-sporting director Iain Moody scuppered the deal.
According to the London Evening Standard, Palace will approach Moyes regarding the position, but the Scotsman is expected to turn down a move to Selhurst Park.
Moyes has been out of work since he was sacked by United in April following a disastrous campaign, which resulted in the Red Devils falling from top spot to seventh in the first season after Sir Alex Ferguson's retirement.
Reports have claimed that Moyes is interested in coaching abroad and is reluctant to return to the Premier League following his stint at Old Trafford.
Former Celtic manager Neil Lennon is currently the bookmakers' favourite to take the job at Palace.