West Ham want to sign Gallagher, who is also subject to strong interest from Tottenham Hotspur, and it has emerged they lodged a bid with Chelsea for him.
The proposal put forward by West Ham came in at £37m plus add-ons, an amount West Ham feel is fair.
However, Chelsea have rejected the proposal as they believe that Gallagher is worth more money.
That stance has led to a standoff as West Ham are unwilling to improve their offer for the midfielder.
West Ham have brought in £100m from selling Declan Rice to Arsenal, but boss David Moyes wants two midfielders, while further additions may also be needed.
It remains to be seen if either Chelsea or West Ham change their positions regarding Gallagher.
All eyes will also be on how Tottenham react to the situation, with Spurs admirers of Gallagher and holding an interest in taking him to the Tottenham Hotspur Stadium this summer.
