I don't believe any of the tabloid gossip until I see a player holding up a jersey. And even then I'm skeptical!
Other than Pulisic, I wouldn't expect any player to be able to just jump right into Klopp's system to replace Mane. I think that we are looking at Sturridge, Origi, Coutinho, Firmino and perhaps Lallana occupying those top 3/4 forward positions. The timing of Henderson's injury is particularly problematic in that Lallana could have otherwise proven a reasonable solution (with Can, Henderson and Windaljum at mid). But I don't know that it works if they have to play Stewart or Lucas (if he stays).
At any rate, with Pulisic seeming highly improbable, I'd be any January signings are long-term prospects. So much so that I would't be at all surprised to see someone like Forsberg get loaned back.