Chelsea and Real, meanwhile, had marched ahead in their pursuits of Alisson, with £53m (€60m) and £44m (€50m) bids lodged respectively.
According to Italian newspaper Leggo, Liverpool have revived their interest in getting Alisson on-board this summer.
They say Liverpool have trumped Chelsea and Real by proposing a deal worth £57m (€65m).
However, that figure is still some way short of being high enough to convince Roma to sell.
Leggo indicate director of football Monchi and chairman James Pallotta will only do business for £66m (€75m).
And that’s because they want to spend up to £22m (€25m) on Paris Saint-Germain’s Alphonse Areola, with the rest to be splashed out on a forward-thinking player.