The east London club would prefer to keep Rice but there is an acknowledgement his situation is reaching a critical point given he effectively has two years remaining on his contract at the London Stadium.
The 24-year-old's deal expires in 2024 but there is an option to extend by a further 12 months.
Sources have told ESPN that Arsenal want to pursue a deal for Rice at the end of the season while Chelsea, Manchester City, Manchester United, Newcastle United and Liverpool are monitoring developments.
Arsenal are keen on bringing in two central midfielders this summer with Rice and Brighton's Moises Caicedo on their shortlist.
Chelsea have long held an interest in Rice -- who was released by the club aged 14 -- but their willingness to pursue a deal will depend on the identity of their new head coach and how quickly they can trim down a bloated first-team squad.
City want to strengthen their midfield but a move for Rice will likely be influenced by whether or not they can win the race to sign Jude Bellingham, although Real Madrid are confident of landing the Borussia Dortmund midfielder.
United's priority is to sign a top striker with Tottenham's Harry Kane under consideration. They would likely need to raise further funds to sign Rice, although it remains to be seen how the sale of the club affects their summer plans with United boss Erik ten Hag admitting earlier this week he was unclear how much money they will have to spend amid ongoing takeover talks.
Newcastle will seek to invest more aggressively should they qualify for next season's Champions League, as appears increasingly likely, while Liverpool are reportedly considering joining the battle for Rice after pulling out of a move to sign Bellingham.
Although West Ham hope their valuation will prevent a summer departure, sources have told ESPN that Rice previously rejected multiple offers to extend his contract on improved terms.
Sources added that Rice also made little secret of his desire to compete for trophies at the highest level.
West Ham face a Europa Conference League semifinal against AZ Alkmaar later this month but sit 15th in the Premier League and remain in danger of being pulled into a relegation battle.
"100% I want to play in the Champions League," he said. "For the last two or three years I've been saying that.
"I've been playing consistently well for my club and I feel like I really want to keep pushing. I see my friends here who are playing Champions League and for big trophies.
"You only get one career and at the end you want to look back at what you've won and the biggest games you've played in."