The respective valuations has increased confidence at Liverpool that the 25-year-old will be their new first choice goalkeeper next season.
However, Roma are thought to be holding out for a further €5 million for the former Internacional stopper, while the Reds also face competition from Real Madrid and Chelsea for the Brazil worldwide. Real Madrid, another possible suitor, have yet to make contact with Roma.
Should the Brazilian global make the move to the Premier League he would overtake Italian Gianluigi Buffon as the most expensive goalkeeper of all time.
Gianluca Di Marzio has reported that the Brazilian goalkeeper is closing in on a move from Roma to Liverpool.
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Given the amount of money Liverpool splashed on the likes of Virgil van Dijk, Naby Keita and Fabinho, an extra €5m on top of their original offer will surely be a formality.
Sky sources had previously indicated Liverpool had no plans to bid for Alisson this summer, but that was earlier in the window when the Premier League believed the Brazil global was going to Real Madrid. Tentative inquiries in January were also dismissed.
While Liverpool fans are fiercely loyal to manager Klopp, his eagerness to stand by Karius has been a point of major concern in recent weeks, with Anfield goal-scoring great John Aldridge among those voicing his assertion that the Reds need a new No.1.
The Roma stopper had a distribution accuracy of 83% per 90 last term, a clear upgrade on both Karius (72%) and Simon Mignolet (75%) respectively - and he even registered an assist in Serie A last term.