The 23-year-old has established himself as a prominent figure at Valencia since joining the club in 2021, keeping 23 clean sheets in 79 appearances across all competitions.
Mamardashvili is understood to have earned admirers from across Europe with his performances between the sticks for Los Che, who currently sit 11th in the La Liga table.
The 6ft 6in Georgia international is under contract with Valencia until June 2027, but his long-term future at the club is believed to be uncertain due to their financial predicament.
According to Fichajes, Valencia are 'immersed in considerable debt' and will be required to sell first-team players to balance their books.
The report adds that one of the standout names who could depart is Mamardashvili, with Valencia's financial situation keeping Premier League clubs in 'suspense'.
Man United, Tottenham and Newcastle are all said to be among the English top-flight clubs who hold an interest in Mamardashvili, with all three teams looking to strengthen in the goalkeeping department ahead of next season.
Newcastle could even bolster their goalkeeping ranks as early as January as their first-choice shot-stopper Nick Pope is facing four to five months on the sidelines with a dislocated shoulder.
Martin Dubravka has since deputised as the Magpies number one in Pope's absence, while Loris Karius and Mark Gillespie are two other goalkeepers at Eddie Howe's disposal.
Both Man United and Tottenham signed new goalkeepers from Serie A in the summer, with the Red Devils recruiting Andre Onana from Inter Milan and Spurs luring Guglielmo Vicario from Empoli.
Onana, who moved to Old Trafford following the exit of David de Gea, has failed to reach the heights expected at Man United this term and has been criticised after making several mistakes leading to goals.
Erik ten Hag is expected to stick with the Cameroonian, who is under contract until June 2028, for the time being, while Altay Bayindir - another summer signing - and Tom Heaton are two backup options available to the Man United manager.
Vicario, meanwhile, has made an impressive start to life at Spurs, starting all 16 Premier League matches and boasting a 71% save percentage.
The 23-year-old Italian has seamlessly replaced Hugo Lloris, who has been frozen out of the Tottenham first-team fold by Ange Postecoglou after expressing his desire to seek pastures new in the summer.
Fraser Forster is acting as Vicario's understudy and the 35-year-old recently committed his future to Spurs by penning a contract extension until 2025.