The attempts to lure Borini to the Stadium of Light have been one of the longer-running sagas of this summer's transfer window after the Italian had a successful time on Wearside last season.
Borini is believed to want to fight for his place in the Liverpool team with the club back in the Champions League this season though his opportunities may be limited by Daniel Sturridge and Rickie Lambert, along with potential new signing Mario Balotelli.
However, Goal are reporting that Serie A side Napoli could be interested in the player and are lining up a bid to sign the forward just before the deadline.
There have been several rumours about Borini potentially returning to Italy with Inter Milan also linked.
If Napoli's interest materialises into a formal offer, the attraction of returning to his home country could appeal to Borini, though the prospect of playing Champions League looks precarious after the side's 1-1 draw with Athletic Bilbao in the first leg of their play-off game on Tuesday night.
Liverpool seem to have been willing to let the striker leave the club for the right price, though it was thought earlier on this summer that Brendan Rodgers felt he needed four strikers to deal with the demands of more Champions League action.