Roma want Liverpool to raise their offer for goalkeeper Alisson to €80m, while Chelsea are soon expected to enter the race.
It had seemed the Brazilian would be heading to Real Madrid this summer, but Los Blancos are not prepared to pay the figures which Roma have been demanding.
Liverpool have taken the opportunity to press the accelerator and are prepared to pay €70m plus bonuses.
But, according to Italian daily Il Tempo, Roma are asking Liverpool to increase their offer to a fixed €80m.
It is claimed that a proposal of €75m might be enough to convince Roma to sell.
The Reds may want to get the deal done quickly as Chelsea are soon expected to join the race.
New Chelsea boss Maurizio Sarri is keen for the Blues to land Alisson, though the club have first been waiting to see what happens with Thibaut Courtois.
The Belgium international is into the final 12 months of his contract and Real Madrid have been showing interest in him.
