Barcelona and Liverpool are negotiating a deal for Coutinho, who wants to join the La Liga club.
Coutinho, 25, has handed in a transfer request and is yet to play for Liverpool this season.
And Spanish radio show Onda Cero claim time is running out for Barca to settle on a transfer fee.
According to the report, Liverpool have set Barcelona a Monday ultimatum to match their valuation of Coutinho.
The Reds have previously said they won’t sell their Brazilian gem but appear to have softened that stance in recent days.
Three bids – the last worth £118million – have been rejected by the Anfield side.
Owners Fenway Sports Group hold the final decision on the move and Onda Cero claim the Monday deadline is absolute.
Liverpool would want to sign a replacement for Coutinho if they sold him and need time to secure a move with Thursday’s transfer deadline looming.
They have already failed to sign RB Leipzig’s Naby Keita and have moved on to Bayern Munich youngster Renato Sanches.
Barca expect to complete the signing of Ousmane Dembele from Barcelona on Monday but that deal will not prevent them from continuing their pursuit of Coutinho.