The 24-year-old was heavily linked with a move away from the Hammers last year but ultimately remained at the London Stadium, and he has again been a vital player for David Moyes's side during the 2022-23 campaign, scoring twice and registering four assists in 34 appearances.
West Ham can extend Rice's deal for another year until June 2025, but there is a growing feeling that the capital giants will reluctantly accept selling their captain in the upcoming market.
Manchester United, Manchester City and Chelsea are all believed to be keen on the England international, but Arsenal's interest has grown over the last few months.
According to the Evening Standard, the Premier League leaders are confident that they are in pole position to secure his signature ahead of the 2023-24 campaign.
The report claims that Rice is Arsenal's number one transfer target this summer, and the club's owners are prepared to sanction a move that could be close to £100m.
West Ham sporting director Mark Noble recently admitted that Rice would likely leave the club in the near future in order to realise his dream of playing Champions League football.
Speaking during the 2022 World Cup in Qatar, Rice also hinted that he would be looking to depart the London Stadium in the not too distant future.
"One hundred per cent I want to play in the Champions League," Rice told reporters at the time. "For the last two or three years I've been saying that.
"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. I really want to do 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."
Arsenal are not mathematically guaranteed Champions League football for next season, but they are 22 points ahead of fifth-placed Newcastle United and eight clear of second-placed Manchester City in the Premier League table, with the Gunners fighting to win the title.
Rice has gained European experience during his time at West Ham, featuring in the Europa League and Europa Conference League.
The Hammers have struggled in the Premier League this season, though, only picking up 24 points from 26 matches, which has left them in 18th position heading into the next set of fixtures.