Senegal international Mane pulled up in the first half of Liverpool's 2-1 win at Wolves with a hamstring complaint.
The 27 year-old was absent for the 2-2 draw with Shrewsbury Town in the FA Cup fourth round and will miss the Premier League trip to West Ham midweek.
However, Klopp only expects Mane to miss one other match - a home clash against his former club Southampton on Saturday - as the damage is not as bad as first feared.
"Sadio will not be involved for West Ham and probably not Southampton," Klopp said. "Afterwards he should be fine, then the week after that should be perfect for him to get back on track.
"He has a small muscle tear, which is serious enough to rule him out, but we've been a little bit lucky with the injury as well so it's not that serious.
"But when you have three games in a week it means he's out for three games."
Mane has started 21 of Liverpool's 23 Premier League matches this season and has been used 31 times in all competitions.