The Red Devils want to add a new first team shot stopper to Erik ten Hag's squad in the summer.
United could release veteran David De Gea, as he has rejected their most recent contract proposal.
Per Cope in Spain, Mamardashvili is emerging as a serious alternative for United.
The 22-year-old has only managed five clean sheets for Valencia this term.
However, he has impressed in La Liga, despite the club being in relegation trouble.
He is good with his feet and appears to be a keeper that can thrive in the modern Premier League.