The attacking midfielder is currently available on a free transfer after his contract with the French side expired at the end of last season, the Daily Express report.
The 29-year-old made just 19 appearances in his injury-laden final campaign, eight of which were starts.
However, despite the fitness problems and lack of game time, it seems that, according to the Express, the Premier League trio are still interested in his signature.
Brendan Rodgers is looking to get ahead of his English rivals in the race to sign the France international, and add to the six players already lured to Liverpool this summer.
James Milner, Danny Ings, Adam Bogdan, Joe Gomez, Nathaniel Clyne, and Roberto Firmino have all been signed [Sky Sports], while Aston Villa ace Christian Benteke could make it number seven, as he is set to undergo his Merseyside medical on Monday [talkSPORT].
Should Gourcuff arrive in Merseyside, he would add to the already burgeoning ranks of attacking midfielders at Anfield.
Recently signed Firmino joins Adam Lallana and Philippe Coutinho as the players competing for the position, and as a result the former Bordeaux man – in reality – may not actually be a needed signing for the Reds.
However, further reinforcements are likely still on the agenda for Liverpool, with improvements needed after finishing sixth in the Premier League last time round.
A return to the Champions League will be one of the Reds’ targets, and with all those around them also strengthening – it will be a tough ask.