With Chelsea on the verge of signing Tyler Adams and still chasing Moises Caicedo and Romeo Lavia, the club are keen to move on Gallagher.
The Chelsea academy product has impressed in pre-season but the club are inviting offers from Premier League clubs for him.
West Ham failed with a bid for him earlier in the window and have moved on to other targets for the moment.
But it has been claimed that Gallagher has continued to remain an option for Tottenham this summer.
He is on Spurs’ shortlist and the club are closely monitoring his situation at Chelsea in the ongoing window.
They could enter the race to take him to north London with less than three weeks left in the window.
Chelsea are prepared to cash in on him and want somewhere around £45m from his departure this summer.
