The future of the 24-year-old has been a constant topic of debate with Blues co-owners Todd Boehly and Clearlake Capital allegedly against handing the academy graduate a new contract
Their preference is to cash in on the homegrown talent for a considerable profit, with the likes of Spurs and Aston Villa having been heavily linked with a big-money move.
However, according to the London Evening Standard, Spurs are of the opinion that Gallagher is unlikely to accept any proposal that they put his way.
If Chelsea were to benefit most from a sale, it would need to be finalised by June 30 in order to bolster their books ahead of the next accounting date.
Gallagher has already stated that his full focus is on representing England at Euro 2024, with the Three Lions now certain to make the last 16 and remain at the tournament past June 29.
Across London, West Ham United may reportedly have to fend off strong interest from Saudi Arabia for Mohammed Kudus.
In his first season at the London Stadium, Kudus contributed 14 goals and six assists in all competitions, making an instant impact after a high-profile move from Ajax.
Nevertheless, with the Hammers having missed out on European football for 2024-25 and now having a different manager at the helm, interested clubs may sense an opportunity to sign the Ghana international.
As per ESPN, Al Ittihad are weighing up the possibility of trying to sign Kudus during the summer transfer window.
Any offer would need to exceed the £38m that West Ham paid less than 12 months ago, with the 23-year-old also on a long-term contract.
Al Ittihad are looking to rebuild after a dismal defence of their Saudi Pro League title, finishing 42 points adrift of eventual winners Al Hilal.
Over in the Netherlands, Ajax would allegedly like to secure the services of Nice defender Pablo Rosario.
The Dutch giants are looking to transform their squad after a poor 2023-24, and they now have Francesco Farioli - formerly of Nice - in charge.
According to transfer expert Fabrizio Romano, Rosario has been identified as one of Farioli's priority signings as he bids to make his mark in Amsterdam.
While an agreement is expected between the clubs, it is claimed that Ajax and Rosario are yet to find common ground over personal terms.
Rosario was born in Amsterdam and has previously represented rivals PSV Eindhoven, but the 27-year-old is settled at Nice having made 110 appearances across three seasons.