The duo suffered the injuried during the 4-2 win over Derby County in the Carling Cup on Tuesday night.
“We sent Anderson for a scan and there’s no break,” said Sir Alex Ferguson. “But there was some ligament damage. We think he’ll be out for three weeks.”
Nani also suffered an injury in the game, but will only be out for a week.
Ferguson added: “Nani suffered a groin injury and he’ll be out for a week. Jonny Evans will be out for three weeks.”
The United gaffer is struggling with several injuries ahead of their FA Cup fourth-round tie against Tottenham on Saturday.
He continued: “The ones who are out are Rio Ferdinand, Anderson, Nani, Rafael, Patrice Evra, Wayne Rooney, Jonny Evans and Ji-sung Park [hamstring injury].”