Barcelona had a bid of around £90million for the Brazil international rejected by the Reds on Wednesday.
And now Coutinho has declared to Liverpool his desire to leave - news that emerged on Friday only hours after the club's owners had issued a statement stressing their stance that the 25-year-old would not be sold during the current transfer window.
Coutinho is understood to have submitted the transfer request personally via email.
Barca, who sold Neymar to Paris St Germain last week for a world-record £200.6m, made their offer on Wednesday having already had a £72m bid for Coutinho turned down in July.
Liverpool boss Jurgen Klopp has gone on record a number of times emphasising the player is not for sale, and the club's owners Fenway Sports Group (FSG) backed that up on Friday morning by saying in a 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 (transfer) window closes."
That message appeared on Liverpool's official website just prior to Klopp speaking at a press conference, during which he said of FSG's statement: "I think if somebody, in this case FSG, is saying something like this, that is the statement.
"I have said what the club's decision is about this and now you can read it - that is maybe the only difference. And that is the way it will go."
Soon after the press conference, reports emerged that Coutinho had handed in a transfer request, and although initially Liverpool insisted that was not the case, it is understood they have now received one from the player.
Also on Friday, a statement released to Sky Sports News, which was said to be from one of Coutinho's close family members, was read out on the channel.
The statement said: "Philippe has tried very hard to find an amicable solution to this situation but to no avail.
"He has tremendous love for the club and its fans but, like Steven Gerrard and Luis Suarez have pointed out in the past, Liverpool does not let its players leave on amicable terms."
In the press conference ahead of Saturday's Premier League clash with Watford - a game Coutinho will miss due to a back problem - Klopp was asked about trying to keep the former Inter Milan man happy.
The German said: "I can't keep players happy. That's how life is. It's not about this at the moment. Phil is not available for us and that for me is the bigger issue. All the rest is just between me and all the boys, not just Phil."
Klopp also said he had "nothing to say about this" when asked what Coutinho had said to him regarding the Barcelona interest.
As well as being unavailable for Saturday's contest at Vicarage Road, Coutinho is a doubt for Tuesday's Champions League qualifying play-off first leg against Hoffenheim in Germany.
Coutinho, who joined Liverpool from Inter for £8.5m in 2013, signed a new five-year deal with the Merseyside outfit in January that did not include a buy-out clause.
He was their leading scorer with 14 goals in all competitions last term, a season that saw Klopp's men secure a fourth-placed finish in the Premier League.