Jurgen Klopp is considering a £22.5m move for Germany international Jonas Hector.
The Reds have also lodged a £33.5m bid for Joao Mario, according to the player’s father.
Starsport understands United had been keen on the Sporting Lisbon star but now Liverpool have waded into the race.
Mario’s dad said: “"I was informed that the club said no to €35m plus €10m in bonuses from Inter and to €40m from Liverpool.”
Mario Gotze had been the Reds’ top target but he’s rejoined Borussia Dortmund.
Klopp still wants a left-back to challenge Alberto Moreno - despite his public insistence he might not make any more signings.
Asked if he anticipates more additions, Klopp told ESPN: "If I had to guess I would say rather no, but I don't know actually because it's a long time until the 31st of August and I have no idea."
Sadio Mane Mane rejected Manchester United to join Liverpool for £34m from Southampton.
He is joined by Georginio Wijnaldum, a £25m buy from Newcastle.
Loris Karius arrives for £4.7m from Mainz and he’s expected to take the No.1 shirt from Simon Mignolet.
Ragnar Klavan, Joel Matip and Marko Grujic have also signed on at Anfield.
A number of young players have signed new deals.
Many of the ‘outs’ have already been done - although a few more are expected to leave on loan.
Mario Balotelli could be one of them with Fenerbahce, Napoli, Ajax and Besiktas showing an interest in the flop striker.
Lucas Leiva also looks likely to leave after Galatasaray came to an agreement with the Kop for his transfer.
A few youngsters are likely to depart too as Klopp softens his stance on keeping them all at the club.
The biggest name departures include Joe Allen - who has joined Stoke for £13m - and Jordon Ibe. The speedster has joined Bournemouth for £15m.
Martin Skrtel, Kolo Toure and Jerome Sinclair have all left the club too while Brad Smith will link up with Ibe for the Cherries.
