Discussing what looks set to become one of the summer's longest-running and most high-profile transfer sagas on Monday night (31 July), French radio station RMC, as relayed by Marca, said that an initial €80m (£71.5m, $94.5m) approach had been rebuffed and declared that an improved follow-up bid worth €100m (£89.3m) had also been turned down.
However, it was further suggested that Liverpool are expected to eventually sell for a higher price despite their firm public insistence that the Brazilian international is simply not for sale.
Links between Barcelona and the versatile Coutinho are certainly nothing new and have really grown in intensity over recent weeks.
The Catalan giants are eager both to source a long-term replacement for midfielder Andres Iniesta and also begin to fill the sizable void in their attack that will be left by Neymar's imminent world-record move to Paris Saint-Germain.
Liverpool are understood to have slapped a mammoth £133.7m price tag on Coutinho after news of that opening offer became public knowledge, although their attempts to keep the player, who only signed a new five-year deal worth £8m-a-season in January, may be undermined by reports that he has already agreed personal terms with Barcelona and has made no secret behind the scenes of his desire to move on.
Jurgen Klopp has reiterated on several occasions that Liverpool have absolutely no intention of cashing in on Coutinho, who was handed the captain's armband for a 3-0 pre-season friendly victory over Hertha Berlin at the weekend.
Addressing the situation again yesterday during a pre-Audi Cup press conference held in Munich, he said: "If you say he is not available to be sold, where is the interpretation? The word 'not' means there is no interpretation. There is nothing new."
Klopp, who has bolstered his squad over recent weeks with the signings of Mohamed Salah, Andrew Robertson and Dominic Solanke while being met with frustration in pursuit of Virgil van Dijk and Naby Keita, has declared that Liverpool were not a "selling club". He also described it as perhaps the "best news" of the summer that they had not yet and would not be parting with any key players that they wished to keep.
Barcelona have made only three relatively low-key additions during the current transfer window to date, with Portuguese right-back Nelson Semedo joining the returning Gerard Deulofeu and young defender Marlon Santos.
PSG stalwart Marco Verratti was their initial priority midfield target before focus switched back to Coutinho, while the club are still keen on bringing former Tottenham Hotspur flop Paulinho back to Europe from Chinese outfit Guangzhou Evergrande.
Meanwhile, the likes of highly-rated teenage USA winger Christian Pulisic and prominent Arsenal target Thomas Lemar are believed to be among the potential options for Liverpool should they choose to alter their hardline stance on Coutinho.