Coutinho stated his desire to leave via email. The club rejected the request.
On Wednesday, the Reds rejected a 100 million euros bid (£90m) from Barcelona for the 25-year-old, who is out of Saturday's trip to Watford with injury.
In a statement on Friday, the club's owners said its "definitive stance" was that "no offers will be considered".
Speaking at an earlier news conference previewing the league match at Vicarage Road (12:30 BST kick-off), Liverpool boss Jurgen Klopp said the Brazilian would miss the season opener because of a back problem.
Klopp said he was also a doubt for Tuesday's Champions League play-off first leg at Hoffenheim as he had "not been in training since last Friday".