The highly-rated French midfielder has turned down the chance to extend his deal with the Ligue 1 champions and will leave the club this summer, although he is yet to agree terms elsewhere despite some of Europe's top clubs being interested in him.
Barcelona were thought to be leading the race for his signature, but they have since cooled their pursuit having already secured a deal to sign Frenkie de Jong from Ajax.
Tottenham Hotspur and Manchester City have also been credited with an interest in Rabiot, but Paris United claims that Liverpool have made the most recent approach.
Talks are understood to be ongoing with all three Premier League clubs, while there is also interest from German giants Bayern Munich and Borussia Dortmund.
However, Rabiot's wage demands could prove to be a sticking point in negotiations, with the 23-year-old demanding a salary of around €10m (£8.6m) after tax in addition to a signing-on fee of between €10m and €15m (£8.6m-£12.9m).