According to the Sun, the clubs are considering a move for the 21-year-old, who has one year remaining on his current deal, after a reported contract stand-off has unsettled the Republic of Ireland international following his breakthrough season with the London club.
Rice made 34 appearances across all competitions last season, making his full international debut for the Republic of Ireland in March, and was expected to push on and become a key player for the club in coming seasons.
However news of contract disputes, coupled with West Ham announcing the signing of France Under-21 international centre back Issa Diop this week, indicate Rice's future under Manuel Pellegrini is becoming more and more uncertain by the day, and the Hammers' Premier League rivals in the form of Palace and Bournemouth hope to capitalise on the situation.
The signing of Diop is expected to be far from the last deal West Ham conclude this summer. Already in the door are right back Ryan Fredericks and Poland goalkeeper Lukasz Fabianski as well as Diop.
And the club are reported to be in advanced talks with Lazio over Brazilian winger Felipe Anderson among a list of other targets as Pellegrini looks to overhaul his squad before his first season in charge gets underway in August.
Palace, meanwhile, may look to Rice, who can operate as a holding midfielder, as an alternative to former Arsenal midfielder Jack Wilshere, who looks set to snub them in favour of a move to the Hammers.