The Gunners have long-admired the 24-year-old and Standard Sport understand they have now moved ahead of both Manchester clubs in the race to sign him.
Mikel Arteta has made the West Ham midfielder a major summer target in the belief his arrival could help take the club to the next level.
Any deal, which is likely to cost in the region of £100m, would smash Arsenal’s transfer record, which currently stands at the £72m paid for Nicolas Pepe in the summer of 2019.
Rice spoke of his desire to play Champions League football while away on World Cup duty with England and West Ham are increasingly resigned to losing their academy graduate.
The midfielder – whose contract runs until the end of next season - has rejected several offers of a new deal at the London Stadium and the club have privately accepted they will have to sell this summer or else risk missing out on a major transfer fee.
While West Ham do have the option of a 12-month extension, this summer surely represents the last time to sell for anything approaching their huge demands.
Chelsea have been linked with a move for Rice but it is understood their interest has cooled.
Speaking after West Ham’s loss at the Emirates Stadium in December, the England star said: “They are a top team you can see they are coached incredibly well, have a lot of attacking talent, a young squad and they can go all the way for sure.
“We tried as hard as we could to stop them, we came up short. You can’t sustain that for 90 minutes against a team like Arsenal, it’s the same against City. With their free-flowing football they managed to get through.”