He is said to have a £42m release clause in his current deal, which Liverpool are understood to be ready to activate.
That is something they should look into as soon as the summer transfer window opens, with midfield arguably the weakest area in Klopp’s squad.
Alex Oxlade-Chamberlain has failed to stamp his mark on the team, with Milner nearing the end of his career.
Liverpool will now have the prime chance to refresh in that area by bringing in Gavi, who has already proven to be capable of starring at the top level.