Bamford, who joined the club in a reported £10million deal in July, sustained a posterior cruciate ligament injury playing in an Under-23 fixture against Bristol City at Elland Road on Saturday (AEST).
The striker has made five substitute appearances in the Championship this season, with Leeds level on points with his old club Middlesbrough at the top of the table.
"Thanks to everyone for the messages of support," Bamford wrote on Twitter. "I'm gutted by the news but I'll be back ready to help the team as soon as possible and I'll be behind the boys all the way in the meantime."
Bamford scored his only Leeds goal to date in an EFL Cup win against Bolton Wanderers last month.
"This is clearly disappointing news for Patrick and for everyone at Leeds United," said director of football Victor Orta.
"Patrick has settled in well, quickly becoming an important player for Marcelo Bielsa and a popular part of the group.
"We have an excellent medical team at the club and Patrick is in the best possible hands, he will be back available to us during an important part of the season and he will still play a key role in our campaign."