Fonte, 33, handed in a transfer request at Southampton earlier this month and was linked with a move to Manchester United in the summer.
The Daily Mirror says Reds boss Jurgen Klopp wants to add more experience to his back four and views Fonte as an ideal candidate to help bolster his side's title challenge.
The Portugal international would command a transfer fee of around £10m ($12m), according to the paper.
Speaking on 5 January, Southampton director of football Les Reed confirmed that Fonte, who has 18 months left on his contract, had asked to leave the club and that there were no official bids for the player as yet.
"He's had several opportunities to improve the situation. He's reserved his right not to do that," Reed told the BBC.
"He's made it very clear he would like to explore the opportunities for a transfer.
"That's where we are at the moment, Jose wants to leave the club. He's formally asked for a transfer."
The Mirror says Liverpool could attempt to use Mamadou Sakho in a swap deal involving Fonte with Southampton.
The 26-year-old France defender has been frozen out of the Reds squad by Klopp after he refused to leave the club last the summer.
Sakho angered Klopp by breaching discipline thrice during Liverpool's pre-season tour of North America – arriving late for a team flight, missing a team meal and skipping a treatment session in California.
He further antagonised the German coach by posting on social media in September to voice his frustrations at being frozen out of the squad.
The France international has not played since being banned for failing a drugs test late last season – a decision that was later reversed by Uefa who cleared him of doping charges.