The Gunners submitted a third bid for the England international late on Tuesday night.
While the Hammers did not accept the offer, it was not immediately turned down.
Instead, the two clubs entered into advanced negotiations to restructure the payment of the £105m offer.
City submitted a £90m bid of their own earlier this week, which was rejected immediately.
Pep Guardiola is keen to bolster his midfield following the departure of Ilkay Gundogan to Barcelona.
But the club will not match Arsenal’s £105m bid as that has gone beyond a figure they are willing to pay.
Mateo Kovacic joined City in a £25m deal from Chelsea on Tuesday and the Croatia international will go some way to replacing Gundogan.
However, City could now enter the race to sign Chelsea and Manchester United target Moises Caicedo from Brighton.
Should Rice join Arsenal, it will be a personal triumphant for Mikel Arteta.
The Spaniard has laid out significant groundwork towards signing Rice and made a personal effort during the course of last season to speak to the midfielder over a role at the Emirates.
City’s financial might and greater guarantee of success threatened Arsenal’s early move but the Gunners are now favourites to sign the England ace.
MORE : Aston Villa favourites for long-term Man Utd and Tottenham target Pau Torres
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