Spurs boss Mauricio Pochettino is lining up a move for the Uruguay international and is eager for a quick agreement.
Pochettino is concerned about competition from Valencia, where coach Marcelino is also a fan.
Spurs see Maxi as a replacement for Fernando Llorente, who is set to leave when his contract expires in June.
However, Los Che may have the edge given their good relationship with Celta. Valencia have taken several players from Celta in recent seasons.