Sissoko, for the first time, has openly admitted he wants to leave the relegated club this summer.
United have slapped a £35million price tag on the midfielder, who impressed playing for France at Euro 2016.
But Sissoko – who struggled for form last season and came under fire from fans – has suggested Newcastle will settle for a lower fee.
A "representative" of Sissoko has also been quoted in France calling for the club to be "reasonable" in its valuation of the player.
Sissoko told L'Equipe: "Newcastle would ask for £35 million? It could be £20million – for me, it changes nothing. The interested clubs will know how to find common ground. I hope that it will be as early as possible."
Sissoko last night posted an emotional message on his social media accounts in the wake of France's 1-0 defeat to Portugal in Sunday's Euro 2016 final.
The 26-year-old thanked Newcastle for allowing him to "progress and discover the English league" and also praised the club's "extraordinary fans".
Liverpool, Arsenal and Tottenham Hotspur have all been linked with moves for Sissoko, signed in January 2013 from Toulouse.