Roma are hoping to up Alisson's release clause to £80million to ward off interest from Liverpool, Paris Saint-Germain and Real Madrid.
The Brazilian goalkeeper is thought to be one of Liverpool's top targets this summer, with the jury still out on both Loris Karius and Simon Mignolet.
It has been suggested Alisson is being considered as a potential long-term number one for the Merseyside club following some impressive performances in Serie A.
But fellow European heavyweights PSG and Real Madrid are also monitoring his situation in the Italian capital closely ahead of the summer transfer window.