The Serie A leaders, who nervously beat Cesena 2-0 on Sunday with a makeshift team to stay four points clear, have suddenly been struck by a raft of injuries with Gennaro Gattuso and Clarence Seedorf also ruled out for around three weeks.
The trio join midfielders Kevin-Prince Boateng, Andrea Pirlo and full back Gianluca Zambrotta already on the sidelines ahead of the Spurs game on February 15.
"In agreement with the player, it has been decided to try an aggressive recovery programme to allow Nesta's return to action in around three-four weeks," a club statement said.
"Surgery will be undertaken, if necessary, at the end of the season."
Nesta, who had a serious back problem two years ago which almost ended his career, suffered the injury against Cesena while Gattuso pulled up in the warm-up with a thigh problem.
Seedorf has a similar injury to Gattuso.
Milan were forced to play youngster Alexander Merkel in midfield alongside usual centre back Thiago Silva on Sunday but did sign Ajax utility player Urby Emanuelson over the weekend.
Antonio Cassano also arrived at the start of the transfer window from Sampdoria, replacing the departed Ronaldinho, while media reports say the Rossoneri still want Genoa's Italy left-back Domenico Criscito.
