The Hammers are refusing to raise their offer past an initial fee of £37m plus £3m in add-ons for the Chelsea academy graduate, who is believed to be surplus to requirements.
The 23-year-old also has interest from Tottenham and would be open to leaving Stamford Bridge having been offered out to clubs for the last six months.
Everton proposed a £45m switch in January, which was accepted by Chelsea but Gallagher was unwilling to move to Goodison Park.
The England midfielder has just two years left on his contract and hasn’t been offered a new deal.
Chelsea want to replace Gallagher with Brighton midfielder Moises Caicedo but have struggled to make progress in talks.
Brighton believe Caicedo’s value is similar to that of Declan Rice, who West Ham sold to Arsenal for £105m and are now trying to replace with two midfielders, including Gallagher.
The Blues are open to signing more than one midfielder next month with Ajax’s Mohammed Kudus recently sounded out.
Celta Vigo’s Gabri Veiga, Everton’s Amadou Onana and Southampton’s Romeo Lavia are also on the shortlist.