The report claims that Palace boss Alan Pardew has been a long-time admirer of the talented central defender, and he is all but set to sign him after the two clubs agreed on a fee.
The Eagles have been intent on strengthening their squad this summer, and the opening of the transfer window was met with the signings of winger Andros Townsend and goalkeeper Steve Mandanda.
While a new striker is now high on the list of priorities, the Eagles have also made no secret of their attempts to bolster the defence as they look for a long-term replacement for the ageing Damien Delaney.
Tomkins, a West Ham youth product, established himself as a first-team regular under former boss Sam Allardyce, but last term he struggled for opportunities as Slaven Bilic opted for Winston Reid and Angelo Ogbonna as his regular centre-back pairing.
The versatile 27-year-old made 23 Premier League starts, but only two between early February and the end of the campaign as he suffered with heavy competition for places.
The report goes on to say that the defender returned to pre-season training early on Friday morning, but his stay in East London will seemingly be brief as he is set to move to Selhurst Park.