The Argentina international left the field in the 71st minute after suffering the problem when West Ham full-back Vladimir Coufal (31) fell on his knee after a challenge.
"Lisandro Martinez has sustained an injury to the medial collateral ligament in his knee and is expected to be out of action for at least eight weeks," United said in a statement.
Martinez was missing for nearly four months earlier this season as he had surgery to fix a recurring foot problem first suffered against Sevilla in April. He returned in January.
"It's very bad for him and very bad for the team," manager Erik ten Hag said on Sunday.
"Last season, when he fell out (of the team), I felt our performances dropped a level. We have to avoid that because we have a good squad and centre-backs who can fill in," he added.
United are in sixth spot, eight points behind fourth-placed Aston Villa who they visit on Sunday.
