Manager Nigel Pearson has said he may return to the loan market for experienced cover should the left-back be absent for a significant length of time.
The Foxes do not have a like-for-like replacement for Konchesky, who was forced off just before half-time at the Reebok Stadium on Tuesday night with teenage striker Jeffrey Schlupp filling in on the left side of the back four for the remainder of the 0-0 draw.
"Paul will be out [against Nottingham Forest on Saturday] I would suspect," said Pearson, who signed Manchester United duo Michael Keane and Jesse Lingard this week.
"He felt his hamstring pull so I think it is pretty certain that he will be missing, which is a shame.
"I think, at some point, we will try and add a bit more experience. Maybe with Konchesky being on the sidelines we may need to have a look at the possibility of more experienced players, but they will have to be right for the football club."