Valencia signed the 21-year-old goalkeeper last summer on an initial loan before making the move permanent in the winter transfer window.
Mamardashvili was expected to be a definite second choice this season but went on to make 18 La Liga appearances and pushed Jasper Cillessen hard for a place in the team.
His performances have piqued the interest of several big clubs and Valencia are now at risk of losing the 21-year-old this summer.
And according to Spanish sports daily AS, Manchester United have already made probes into the potential signing of the young goalkeeper.
The Premier League giants have enquired about the financial conditions that would take the Georgian to Old Trafford this summer.
Manchester United are in the market for a goalkeeper as Dean Henderson is expected to move on in the upcoming transfer window.
The Red Devils are looking at Mamardashvili as a potential second choice who could play in the Europa League and cup competitions.
They believe the Valencia goalkeeper could be a good cost-effective option in the summer.