Mundo Deportivo and Sport claim the Reds are "seriously considering" accepting a bid which would include €40m in add-ons, based on the 25-year-old's performance at the Nou Camp.
Barcelona president Josep Maria Bartomeu is reportedly in constant contact with Liverpool's owners in an effort to convince them to part ways with Coutinho.
The reports add that a deal is expected to be completed after Barcelona play Levante on Sunday [7 January].
But the Liverpool Echo understand that Liverpool are yet to receive any bid for Coutinho - since they rejected three offers during the summer – and that the player has dropped no hint over his desire to leave.
Coutinho submitted a transfer request in an effort to force through his exit from Liverpool during the off-season however he is yet to tell Jurgen Klopp of his want to join Barcelona.
Klopp has dodged questions over Coutinho's future at Liverpool and though he is yet to state he is not for sale the German coach has treated enquiries from the media with derision.
His latest whimsical response to fresh speculation over Coutinho's future came after shirts with the player's name on were advised by Nike, Barcelona's kit manufacturer.
"I couldn't be less interested in anything in the world but somebody told me and I thought 'wow'," the ex-Borussia Dortmund boss said after the 2-1 win at Burnley, which Coutinho missed due to injury.
The former Inter Milan schemer is also likely to miss the FA Cup third round tie against Everton on Friday [5 January] due to an ill-timed thigh problem which has only worked to swell speculation over his future.
A back injury and illness delayed his first Liverpool appearance of the season until 16 September - after the summer transfer window had closed – and Coutinho's latest absence has been questioned.