Having swiftly turned down two bids from the Catalan club for the Brazil global - the last of which totalled €100 million - Fenway Sports Group have supported Jurgen Klopp's repeated insistence that the 25-year-old is not for sale this summer at any cost.
Sky Sports News, which reported the news of the transfer request earlier on Friday, had quoted an unnamed member of Coutinho's family as saying: "Philippe has tried very hard to find an amicable solution but to no avail".
Williams, who played and scored in his side's recent friendly defeat against Liverpool, has been linked with the Reds as a potential replacement for Philippe Coutinho.
The manager said, "Liverpool is not a club that has to sell players".
Liverpool head to Vicarage Road to face Watford in their Premier League opener on Saturday, but it's unlikely that Coutinho will feature in Jurgen Klopp's squad.
In the press conference, Klopp was asked about trying to keep Coutinho happy.
Barcelona offered GBP72 million for the player in its first bid, which was rejected.
It added: "Philippe will remain a member of Liverpool Football Club when the summer window closes".
"Barcelona have been on the back foot a bit, they thought PSG could not buy Neymar", Balague told Sky Sports News. No price at which we are ready to give in.
Although five-time European champions, most recently in 2005, the last of Liverpool's 18 English league titles came in 1990 - before the start of the Premier League era.
The 25-year-old star of Liverpool notched 14 goals for the team in last term. 'That is set in stone.