Rice appears to have played his last game for the Hammers, leading them to Europa Conference glory in Prague last week.
Arsenal have viewed the midfielder as their top summer target since the start of the year and are thought to be close to agreeing a £100million deal to take him to the Emirates.
Manchester United also have long-term interest while Bayern Munich have made him a summer transfer target.
Chelsea has been another possible destination with Rice leaving the Blues academy as a 14-year-old.
According to The Sun, Hammers boss Moyes would prefer for the club to do business with Chelsea and make a play to bring Gallagher to east London in a part-exchange deal.
Gallagher is among the host of Chelsea stars who could be sold this summer as the Blues attempt to trim their bloated squad after last season’s dismal 12th place finish.
However, Rice has made it clear he does not want to return to Chelsea and has his heart set on a move to Arsenal.
As a result, Bayern Munich have also ended their interest in the England international, accepting his decision to remain in England.
West Ham have already signalled their intent to allow Rice to leave with the club reportedly targeting Valencia midfielder Yunus Musah to help fill the void.
‘We promised him he could go. He set his heart on going. You can’t ask for a man who has committed more to us this season,’ West Ham chairman David Sullivan said last week.
‘In due course, he has to get on and we have to get a replacement – or several replacements. It’s not something we want to happen. We offered him £200,000-a-week 18 months ago. He turned it down.
‘It’s cost him £10million to stay at West Ham in that time [in lost wages]. And he wants to go. You can’t keep a player who doesn’t want to be there.’
MORE : Sir Keir Starmer wants Arsenal to sign ‘fantastic’ Declan Rice ‘as soon as possible’ and hails Mikel Arteta
MORE : Arsenal will challenge for title next season if they sign Declan Rice or Moises Caicedo, says Andy Cole
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