Philippe Coutinho's agents and relatives are said to have convinced Liverpool's owners Fenway Sports Group to cash in on the Brazil international during the upcoming January transfer window if Barcelona return with an offer worth a total of €150m (£132.5m, $177.6m) with add-ons.
Sport claims that Coutinho's representatives has managed to soften the stance of Liverpool's owners during several meetings held in Boston since the Brazilian's desired move to the Nou Camp collapsed in the summer.
Barcelona had tried to secure the services of the 25-year-old playmaker after identifying him as Ernesto Valverde's top priority.
Coutinho even handed in a transfer request to try to force the move but Liverpool turned down multiples bids to part ways with his services, with the latest said to be worth around €120m including add-ons.
Fenway Sports Group even issued a statement to make it clear that the former Espanyol star was not going to be sold at any price.
"We wish to offer clarity as regards our position on a possible transfer of Philippe Coutinho," read the statement.
"The club's definitive stance is that no offers for Philippe will be considered and he will remain a member of Liverpool Football Club when the summer window closes."
Barcelona still tried to change Liverpool's minds throughout the summer and only abandoned the pursuit on deadline day, with president Josep Maria Bartomeu claiming that the Merseysiders asked them to pay €200m or forget about his signing.
Speculation linking Coutinho with a move to the Nou Camp has continued since then as the La Liga giants failed to sign an alternative to bolster Valverde's midfield.
In October, Barcelona CEO Oscar Grau himself suggested that the club would make a new effort to sign the Liverpool ace in January if Valverde considers his addition a priority.
And Sport believes that when Grau calls Liverpool to negotiate the transfer ahead of January, he will find a "more favourable" response from the Fenway Sports Group.
The Spanish publication claims the Liverpool owners will still ask for a stunning €150m fee to part ways with Coutinho but will now accept the payment divided in various instalments, rather than all up front.
That fee is still far from what Barcelona offered in the summer and is close to the €105m plus €40m in add-ons the Catalans paid to complete the signing of Ousmane Dembele from Borussia Dortmund.
Furthermore, Sport claims on its Wednesday front cover (29 November) that the dressing room has also encouraged Barcelona officials to make an effort to sign Coutinho in the winter window to give them an extra push ahead of the second-half of the season.
Arsenal ace Mesut Ozil has been mentioned as a potential cheaper alternative in recent days but the report says that "only Coutinho has the backing of everyone" in the dressing room.
The Spanish publication claims that Luis Suarez and Paulinho in particular are pressing for the deal to happen as the striker previously played with Coutinho at Anfield and the latter plays with him in the Brazil national team.