The Premier League champions’ first offer is worth an initial £80m with an extra £10m in add-ons, according to The Athletic.
West Ham have been holding out for an offer of £100m, having rejected multiple bids from Arsenal, the best of which was worth up to £90m.
The Gunners were thought to be leading the race to sign Rice, who is said to still be keen on a move to the Emirates Stadium.
But Manchester City’s interest has left Arsenal sweating, though they are now reportedly preparing to respond to City’s advances with a fresh offer themselves.
Rice has emerged as West Ham’s most important player and a key member of Gareth Southgate’s England squad in recent years.
While he has great affection for West Ham and captained the Hammers to their first major trophy in more than 40 years last season, Rice feels the time is right to leave.
West Ham chairman David Sullivan said after the Europa Conference League win that Rice would likely depart this summer, though his next destination remains unclear.
Manchester City would provide Rice with an immediate chance to win domestic and European trophies, while Arsenal hope to emulate City’s success over the next few seasons and can offer the 24-year-old the opportunity to stay in London.
Chelsea, Manchester United and Bayern Munich all expressed interest in signing Rice earlier in the year but the transfer tussle has seemingly been whittled down to Arsenal and Manchester City.
While Rice has starred in east London and for the Three Lions, Graeme Souness has questioned whether he would improve Arsenal, who finished second in the Premier League last season, five points behind Man City.
‘If you’re looking for a top, top midfield player, I don’t believe he has enough cute passing in him to be that,’ Souness told talkSPORT last week.
‘I think he’d be a really good guy, he’d be great to work with, extremely athletic, he senses danger so he’s got that part of a central midfield player’s game, he’s got that.
‘He can get himself into positions where the guys he is playing against that want to find the striker or make a cute pass, he knows that and not everyone can get that.
‘So he’s got that instinctively, but what he doesn’t have is cute and clever passes when he gets into the last third.
‘I think he’s a little bit predictable when he gets on the ball personally.’
Asked if Rice’s game will improve playing at Arsenal, Souness added: ‘He’d make Arsenal better. Would he personally get better? Not sure.’
MORE : Arsenal defender Jakub Kiwior reacts to Jurrien Timber transfer links
MORE : John Barnes gives advice to Declan Rice over Arsenal or Man City move
For more stories like this, check our sport page.
Follow Metro Sport for the latest news on
Facebook, Twitter and Instagram.
Share your story or advertise with us: Whatsapp: +2347068606071 Email: info@newspotng.com
