The 24-year-old has entered the final 18 months of his contract at the London Stadium, and he is widely expected to leave the Hammers during this summer's transfer window.
A host of clubs, including Manchester United, Arsenal and Liverpool, are said to be keen on the England captain, who hinted at a summer exit while addressing his future at the 2022 World Cup.
"One hundred per cent I want to play in the Champions League," Rice told reporters in Qatar. "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. I really want to do that."
According to 90min, there is also genuine interest from Man City, with the Premier League champions preparing to potentially overhaul their midfield ahead of the 2023-24 campaign.
The report claims that Ilkay Gundogan and Bernardo Silva could both move on this summer, with the former out of contract at the end of June and the latter believed to be keen to seek pastures new.
West Ham are believed to have valued Rice in the region of £150m last summer, with Man United said to have failed with an offer in the region of £100m for the midfielder last year.
However, there is a belief that £100m would be enough to sign the Englishman during this summer's transfer window, with West Ham not in a strong position due to his contract situation.
Rice has again been a key player for the Hammers during the 2022-23 campaign, scoring once and providing three assists in 28 appearances in all competitions.
West Ham sporting director Mark Noble recently admitted that the club's star player could depart the London Stadium in the near future for a Champions League team.
"It's just the world we live in. We're West Ham and these players want to go out and play Champions League football - and do you blame them?" Noble told the Evening Standard.
"Dec is no different. Dec will always be hailed at West Ham as a fantastic player and such a top, top person. But I don't think anyone begrudges him wanting to go and win things. We just hope he'll keep performing like he has been over the last month or so.
"Wherever Declan ends up, or however much someone pays for him, whether it's here or anywhere else, he's worth every penny.
"He's said openly he loves playing at West Ham, but he has got ambitions as a player to go and play in the Champions League and win trophies. I don't think any West Han fan can actually have an agenda over that.
"Do you blame him for having really big ambitions? Of course you don't. If he didn't, he probably wouldn't be as good as he is."
Rice has made 223 appearances for West Ham in all competitions, contributing 11 goals and 12 assists in the process, gaining experience in the Europa League and Europa Conference League over the last couple of seasons.