The Premier League duo are said to be plotting a £3 million offer.
The 26-year-old has been at Celtic since 2012 but is reportedly the subject of Premier League interest, with the defender said to be willing to move from the SPL to the English top flight.
Ambrose, however, has been much criticised for his form by supporters this season, although Celtic boss Ronny Deila has kept faith in him throughout the campaign.
Ambrose could be of use to both West Ham and Crystal Palace as his versatility sees him able to play both at centre-back and at right-back.
Another Celtic defender linked with a Premier League move this January has been Virgil van Dijk but the Mirror now claim Ambrose is a target for two top-flight sides.
West Ham and Crystal Palace have both displayed significant room for improvement defensively in recent weeks - but whether the 26-year-old is the solution is another matter.