Coutinho is also keen on a reunion.
But Barca would rather use him as a makeweight to Paris Saint-Germain in a potential deal for Neymar, according to Mundo Deportivo.
The Reds would only be interested in Coutinho for a significantly reduced price on the one they received from Barca 18 months ago.
Neymar is desperate to escape from the French capital and even refused to turn up for pre-season as he looks to force a move back to Barcelona.
Dspite a promising start to life with the Catalan giants, Coutinho endured a difficult campaign in 2018-19.
He scored just five La Liga goals in 33 outings - three fewer than he managed in 18 games in the second half of the previous season.
Sections of the Barca support let their feelings be known towards the end of the term and there have been rumours linking him with an exit.