Liverpool have met the €75 million (£66.9m) asking price for the Brazil international and have been granted permission to speak to the player about a transfer to Anfield.
Should a deal be completed, the fee would be a world-record amount for a goalkeeper, surpassing the €40m (£34.7m) that Manchester City paid Benfica for Ederson in 2017.
Roma and Liverpool both declined to comment when contacted by ESPN FC.
Alisson, 25, has been a long-term target at Liverpool, whose goalkeeping situation has remained unclear after Loris Karius' errors in the Champions League final defeat to Real Madrid in May. Liverpool manager Jurgen Klopp initially backed Karius, but sources said Danny Ward had a realistic chance of securing the Liverpool No. 1 jersey instead.
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