The offer, reportedly £113million rising to £118m with add-ons, was instantly turned down by the Reds, who are adamant the player is not for sale.
Press Association Sport understands privately Liverpool consider the sums being offered irrelevant as they have no intention of allowing the Brazil international to leave this summer.
Owners Fenway Sports Group said as much last week when they issued a statement explicitly stressing the 25-year-old would be at the club beyond the close of the transfer window.
That prompted Coutinho to hand in a transfer request last Friday, but the club remains resolute.
Regardless of that situation, a back injury means Coutinho will not be facing Crystal Palace this weekend.
Liverpool have already beaten Palace this summer, but manager Jurgen Klopp says that result will have no bearing on this weekend’s encounter.
A 2-0 victory over Frank de Boer’s side in Hong Kong set the Reds on the way to winning the Premier League Asia Trophy, but both teams have lost key players since then.
Adam Lallana and Coutinho both featured in the game, while Wilfried Zaha played for Palace, but all will miss the first match of the season at Anfield.
Arsenal have reportedly held talks with Dennis Praet's agent and could make a £22million move for the Sampdoria midfielder this summer.
Gareth Bale could leave Real Madrid this summer despite proving to be the match-winner during Saturday's Champions League final.