Mundo Deportivo claims on its Tuesday [2 January] front cover that "various sources are confident that the transfer can be completed and announced at the beginning of next week" as Jurgen Klopp's side are now "seriously considering" accepting the offer.
Reports in Spain over the summer had tipped Coutinho to make the move to the Nou Camp after the Brazil international handed in a transfer request to try to force the move.
However, Barcelona finally had to abandon the pursuit on deadline day with club president Josep Maria Bartomeu claiming Liverpool asked them to pay €200m.
But there has been growing optimism in Spain that the transfer could finally be completed in January amid suggestions that Liverpool have now soften their stance.
It has been reported by both Mundo Deportivo and Sport that in the summer Liverpool promised Coutinho they will negotiate his transfer to Barcelona in January as long as the player did not refuse to play the Champions League for Jurgen Klopp's side during the first half of the season.
Nike added more fuel to the saga over the weekend after urging Catalans fans to purchase Coutinho's Barcelona shirt in an advert on their official online store.
The speculation intensified on New Year's Day after Coutinho missed Liverpool's 2-1 victory over Burnley due to an ill-timed injury.
Mundo Deportivo claims Barcelona "assure that there is nothing closed yet" but "various sources are confident that the transfer can be completed after the Catalans clash with Levante on Sunday [7 January]".
The Spanish publication claims that Barcelona president Josep Maria Bartomeu is in constant contact with Liverpool owners to convince them to part ways with the Brazilian as the player is also eager to seal the transfer as soon as possible.
Mundo Deportivo says that Barcelona have already told Liverpool that they are ready to pay €110m plus €40m related to Coutinho's achievements at the Nou Camp.
The report adds the deal will therefore be similar to the one which brought Ousmane Dembele from Borussia Dortmund to Ernesto Valverde's side during the last summer's transfer window.
Barcelona paid €105m to sign the France starlet but will also pay €40m related to the number of games Dembele will play for Barcelona and the club achievements in the Champions League.