The 23-year-old has been a standout performer this summer for a Georgia outfit that have reached the last-16 stage of Euro 2024.
The stopper has another three years left to run on his contract with Valencia, but according to Italian journalist Fabrizio Romano, Los Che are willing to let him leave in the current market.
Romano claims that Valencia have placed a valuation of around €35m (£29.6m) on Mamardashvili, although the club are hoping that the overall fee could eventually reach €40m (£33.9m).
The goalkeeper made the move to Los Che from Dinamo Tbilisi in 2021, and he has represented his current side on 100 occasions in all competitions, keeping 31 clean sheets in the process.
Mamardashvili will be bidding to help Georgia overcome Spain to reach the quarter-finals of Euro 2024, with the last-16 contest taking place on Sunday night.
Sheffield United 'agree Shackleton deal'
Meanwhile, according to Football Insider, Sheffield United have come to an agreement with Leeds United midfielder Jamie Shackleton over a move to Bramall Lane.
The 24-year-old featured on 15 occasions for Leeds during the 2023-24 campaign, including 11 outings in the Championship, but his contact is due to expire at the end of the month.
A number of clubs, including Millwall, have been linked with Shackleton, but it is understood that Sheffield United have won the race for his signature.
The Blades are expected to announce the deal on Monday, with Shackleton's contract at Elland Road officially expiring at the end of the weekend.
The midfielder will seemingly leave Leeds having made 94 appearances in all competitions, scoring two goals and registering three assists, with 27 of those outings coming in the Premier League.
Yates 'closing in on Derby loan switch'
Staying in the Championship, Derby County are allegedly close to agreeing a loan deal with Swansea City for forward Jerry Yates.
The 27-year-old arrived at Swansea from Blackpool last summer, and he scored nine times in 44 appearances for the Welsh outfit during the 2023-24 campaign.
Yates managed eight goals in the Championship last term, but according to BBC Sport, he is expected to spend the 2024-25 campaign on loan with Derby.
The Rams, who finished second in League One last season to secure a return to the second tier of English football, are being tipped to announce the deal over the next couple of days.
Yates started his career at Rotherham United before moving to Blackpool in 2020, and he scored 46 goals in 139 appearances for the Seasiders ahead of his departure in the summer of 2023.