The £30 million summer signing was forced to go to Spire Hospital on Wednesday after a collision with a teammate in training at Melwood earlier in the day.
It was first thought that Mane might have dislocated his shoulder, which would have meant a lengthy lay-off, but the forward's problem is merely a muscle injury.
Despite the positivity of the Mane's scan results, the 24-year-old will play no part in the Reds' trip to Burnley this weekend as they look to follow up on their opening Premier League win at Arsenal.
The Senegal international came out of the club's training centre with his arm in a sling, but signed autographs for waiting supporters and needed no assistance driving home.
The club will keep a close eye on Mane's progress over the coming days - but it is hoped he will be back in time to feature away at Tottenham Hotspur on August 27.
Reds hopeful Mane will return later this month
The Reds' No.19 netted a stunning solo goal and impressed throughout from a wide right position, with Klopp forced to come up with a solution in his absence for their trip to newly-promoted Burnley at Turf Moor on Saturday afternoon.
The boss does at least have strikers Divock Origi and Daniel Sturridge in contention, meaning he has the option to change from his 4-3-3 system to accommodate his other attacking options against the Clarets.
Origi came off the bench late on, while Sturridge wasn't even in the matchday 18 due to a slight hip injury. He only returned to training two days earlier and was not match fit enough to be involved.
Former Burnley striker Danny Ings could also make the squad, missing out on their first game to play 90 minutes and score for Liverpool Under-23s against Southampton.
Regardless, not having Mane - who has made a fine early start to his career on Merseyside - will come as a blow after he whetted appetites with his performance at Arsenal.
The fee that Klopp and co. paid for the No.19 was initially questioned after his inconsistency at former club Southampton, but his accomplished start has gone some way to demonstrating just why the German manager was so keen to bring Mane to Anfield.