Pochettino is widely expected to become the next Chelsea manager as the Blues look for a revival next term.
The former Tottenham boss has already started making preparations for the summer and has identified players around whom he wants to build his project.
While 20-year-old Levi Colwill, who is currently on loan at Brighton, is one of those players that has been eyed, Gallagher, who is interesting both Everton and Newcastle, is another.
The academy graduate was one of those that had previously been linked with a move away from Stamford Bridge in the summer as a dressing room clear-out was being planned.
However, that scenario could now change, dealing heavy blows to Everton and Newcastle United’s chances of signing the 23-year-old.
Two other academy graduates, Mason Mount and Reece James, are also been counted upon by the 51-year-old.
It remains to be seen if Chelsea will change their stance if substantial bids come in for Gallagher in the summer.