There were concerns about the French centre-back, who is on loan at Crystal Palace, after he injured his knee and left the pitch on a stretcher during the South London club's 1-0 defeat to Tottenham Hotspur.
Although Sakho - who fell out with Klopp last year and was allowed to join Palace in January - may not play again this season, Klopp said a scan had revealed no serious damage.
"When we saw pics of Mamadou Sakho it looked awful," he said. "But now we know that there are no ligaments involved. I think he's out for a few weeks."
Any long-term injury would prevent Liverpool selling Sakho in the summer, as has been widely speculated. Palace are one of several thought to be interested in signing the player on a permanent deal.
Klopp also said Liverpool's Adam Lallana and Daniel Sturridge had recovered from injury and were in contention to face Watford on Monday.
"Adam is back in team training for the first time. Today we had a really important session, Adam involved, Sturridge involved."
Lallana has not played since injuring his thigh on England duty at the end of last month; Sturridge hurt his hip training ahead of last week's 2-1 defeat to Crystal Palace.
Although Klopp said Thursday night's goalless draw between Liverpool's closest pursuers, Manchester City and United, "didn't change a lot", it means Liverpool could be fifth, outside the Champions League qualification places, by the time they face Watford.
"It's all about us. We don't have 100 percent influence but makes sense to focus on us. I cannot change the results we had. If we win, which will be difficult, we are probably in third again. So that's all we think about.
"We won the last two away games, even if the last result (against Palace) wasn't good but when I watch again it is not easy to create chances against Palace - and we did.
"We have to fight for what we can fight for and that's a Champions League place. It's a very important game. Nobody will feel like it's a friendly."
(Reporting by Neil Robinson; editing by John Stonestreet)
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