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The Reds had a £110m offer rejected in August and since then the 25-year-old has not trained with or played for Newcastle, but in the final hours of the transfer window a deal has been reached to end the impasse.
The striker is set to undergo a medical on Monday, with Newcastle relenting by accepting the player does not want to play for them any longer.
United also announced the signing of striker Nick Woltemade on Saturday.
He has accused the club of breaking a commitment that he could leave if a suitable offer came from a big club – though that now looks to have changed, with a £5m loyalty payment taking the deal up to as high as £130m.
He will become Liverpool’s eighth first-team signing of the summer, marking the second time they have broken their own transfer record following the £110m capture of Florian Wirtz from Bayer Leverkusen.
In all their summer outlay now stands at around £416m.