The 25-year-old has been in high demand this summer after a string of impressive performances for the Serie A side, breaking through into the first team last season after the end of Wojciech Szczesny's two-year loan deal.
Gianluca Di Marzio (or, at least his website) reports that neither of the Premier League clubs interested in Alisson have yet made an official offer to the player or his club, although that could change after Wednesday's meeting if the club show a willingness to sell - or at least soften on their high asking price.
Chelsea are in the market for a new stopper to prepare for the departure of Thibaut Courtois, who has been strongly linked with a move back to Madrid; albeit to Real, rather than Atletico - where he made his name.
Meanwhile, Liverpool are desperately scrambling around to secure any humanoid shape with hands to start in goal for them at home to West Ham on the opening weekend of the Premier League's new season, with Loris Karius now more closely resembling a chocolate eclair than a reliable top flight goalkeeper.
The German's nightmare showing in the Champions League final, which saw him hand two goals to Real Madrid, appears to have knocked the 25-year-old's already shaky confidence - and another howler in a pre-season match against Tranmere will have done little to improve his confidence in himself, or Jurgen Klopp's in him.
Roma, for their part, are expected to turn to either Paris Saint-Germain's Alphonse Areola, whose position has been put in question by the arrival of Gianluigi Buffon, or Copenhagen and Sweden stopper Robin Olsen.