Gallagher has accepted that he may have to leave Chelsea this summer after not being offered a new deal with two years left on his contract.
He has been at Chelsea since he was six, but is now prepared to move on from Stamford Bridge in the ongoing transfer window.
West Ham and Tottenham are interested in securing a deal to sign the Chelsea academy product ahead of the new season.
But the Hammers are now prepared to march ahead of Spurs in the race to sign Gallagher this summer.
West Ham have the proceeds from the Declan Rice sale to invest in their squad and David Moyes wants to sign two midfielders.
Gallagher has emerged as a top target and the Hammers are ready to edge Spurs out of the race to sign him.
West Ham have also been linked with an interest in Denis Zakaria and James Ward-Prowse this summer.
Chelsea are open to offers for the midfielder despite the lack of senior midfield options in their squad at the moment.
