The pair - known to be close friends - could be set for an Anfield reunion having played together at AC Boulogne-Billancourt and for Les Bleus' youth teams.
Reds boss Jurgen Klopp has made bolstering his midfield a top priority after a below-par season saw them finish outside the top four.
And with James Milner, Alex Oxlade-Chamberlain and Naby Keita to leave when their contracts expire at the end of the month, Liverpool know they have to bring in bodies.
talkSPORT understands Brighton star Alexis Mac Allister is set to sign having completed his medical at the club's AXA Training Centre.
The Argentine World Cup winner could be joined by Nice maestro Thuram - son of Juventus legend Lillian - and Borussia Monchengladbach man Kone.
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It was reported last week how Liverpool have eyed up £38million-rated Kone as an alternative to Jude Bellingham.
Real Madrid have reached an agreement with Borussia Dortmund for the England international for an initial £88.5m, which could rise to £115m with add-ons.
Meanwhile L'Equipe have stated that Liverpool are also keen on Thuram as they line up a transfer offer, with Nice willing to sell him for £34.5m.
And amid speculation linking the pair to Merseyside, Thuram's older brother Marcus - a teammate of Kone - posted footage of the duo together on his Instagram story.
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In the clip, they could be seen chatting on the street as they met up in their native country.
Kone has made no secret of his friendship with Thuram, having played with him at AC Boulogne-Billancourt between 2015-2016.
And they two 22-year-old's have also shared the pitch for France's Under-18, Under-19 and Under-21 sides.
The former Toulouse midfielder told French TV earlier this year: "Khephren is a brother by blood.
"He is my football pal. Before the selections, we were always together."