The striker sustained a serious knee injury, which has now been confirmed by club officials.
Bamford suffered a posterior cruciate ligament injury for the under-23s on Friday.
He has appeared as a substitute five times for Leeds this season.
The club are top of the table in the Championship and hoping to gain promotion this season.
"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."