The centre-half, who was forced off at half-time during last week's FA Cup defeat to Wolves with a thigh problem, had been the subject of interest from former Palace manager Sam Allardyce at West Brom, as well as French club Nice.
The Eagles were happy to let Sakho, whose contract is up at the end of the season, leave this month. However, the latest blow for the 30-year-old will now almost certainly scupper any January move.
"I fear it is going to be weeks," said Hodgson. "It was a very serious injury to happen to someone who had worked so hard to get himself back.
"It was very cruel, I thought, to play a first half against Wolves, the first game he has played for us for a long, long time after many injuries, and to get another big injury.
"But he has come back before and I am always hopeful that an injury that could potentially be a long one, won’t be a long one because he will find a way of getting himself back fit again."