Slaven Bilic is a big fan of the 17-year-old centre-back after he impressed during West Ham's pre-season tour of Austria.
The Sun claim that his rapid rise to prominence over recent months has attracted Arsenal who could make a move for him in January.
The reports add that it would take a hefty offer from the Gunners to persuade West Ham into selling their Under-23’s skipper.
Rice has three years left on his current contract with the Hammers.
He was called into Republic of Ireland Under-21’s set up recently and looks set to have a big future in the game.
Arsenal are also reportedly interested in Rice's West Ham team-mate Dimitri Payet.
The 29-year-old was one of the Premier League’s leading lights last season after linking up with the Hammers from Ligue 1 giants Marseille.
And The Sun also claim Gunners boss Arsene Wenger will move for the £125,000-a-week ace in the new year if Slaven Bilic’s side are still embroiled in a relegation scrap.
Arsenal beat West Ham 5-1 yesterday to leave Bilic and co 17th in the table.