The Hammers are also continuing to work on a move for Club Brugge captain Hans Vanaken after having an initial offer worth around £8million rejected.
West Ham are aware that a deal for Gallagher will be tough to complete, with Crystal Palace, Everton and Newcastle all rival suitors for the 22-year-old.
Gallagher, who starred at Crystal Palace last term, is tempted by the idea of another loan away from Stamford Bridge this season as he is eager to make the England squad for this year’s winter World Cup in Qatar.
Moyes wants to bolster his midfield ranks before the market closes on September 1 and bring in a player who can operate as a No8 or No10.
Vanaken remains a key target and West Ham are still in talks with Brugge. The 30-year-old is keen to play in the Premier League and further negotiations are due to take place later on Wednesday.