The Blues have already lost two key midfielders this summer, with N’Golo Kante and Mateo Kovacic departing for pastures new.
With a large squad though and having splashed the cash in recent transfer windows, further departures are expected, with Mason Mount expected to join Manchester United soon and Ruben Loftus-Cheek heading for AC Milan.
Gallagher is a player that Chelsea are open to selling and he has interest from Newcastle United and Tottenham Hotspur.
However, it has been claimed that Pochettino is to be keen on retaining the midfielder at Stamford Bridge for the upcoming season as the Blues looks to get back on track after a dire campaign.
Pochettino would like to have the opportunity to work with Gallagher and develop his game.
The decision though could be taken out of the manager’s hands given Chelsea’s willingness to cash in for the right price.
And it could sting Pochettino if Gallagher is sold to his former club, Tottenham, who will be looking to challenge Chelsea for a top four spot.