Keita was formerly a teammate of Werner's for two years in Leipzig. Mane, meanwhile, was impressed by Werner last season when he began watching Leipzig games in preparation to welcome close friend Keita, who was a colleague of his at Red Bull Salzburg, to Anfield.
Werner is only under contract with Leipzig until 2020 and it could be that he is sold at the end of this season if the player decides he is ready for a fresh start and doesn't agree an extension.
Speaking to the German media this week, via ESPN, Keita said of Werner: "We played really well together at Leipzig, and he was one of my best mates. It's clear: Should he move, then I ask Timo to join us here at Liverpool."
Mane added: "I'd advise him to join Liverpool. I like his style of play. Very forceful, good finish, always giving it 100 percent. If he wants to join us, we'd welcome him to our team.
"When it was clear that Naby would join us, I often watched Leipzig matches. It was dreadful for me at the start. Naby saw quite a few red cards...Naby missed some important games and I could watch other players more closely. And Timo stood out."
Werner will also be known to Jurgen Klopp. Prior to joining the Leipzig revolution in 2016, a teenage Werner played three times for Stuttgart against Klopp's Borussia Dortmund side in the Bundesliga and even registered an assist during a 2-2 draw at Westfalenstadion in 2014.
This season, the 22-year-old has scored 13 times for Leipzig in all competitions and remains on course to reach the 20 goal milestone for the third consecutive campaign.
With little in the way of depth behind starting front three Mane, Mohamed Salah and Roberto Firmino, Liverpool are likely to find themselves in need of fresh attacking options in the summer.
Daniel Sturridge has started just twice in the Premier League so far this season and is out of contract this year, while Dominic Solanke has already left the club. A heroic winner in the Merseyside derby has also failed to propel Divock Origi any closer to a more regular place.