Marca says Real feel that a deal can be closed at €60m.
Thus far, Los Blancos have only made an offer of €50m for the shot-stopper, an offer which Roma have turned down without much consideration.
The Italian club value their first choice at €70m, a fee which has thus far been deemed unreasonable by both Liverpool and the European champions.
Florentino Perez has long been keen to add a new goalkeeper to the squad at the Estadio Santiago Bernabeu and he pushed heavily for Kepa Arrizabalaga to join in January.
Everything is on track with regards to a deal to bring Alisson to the Estadio Santiago Bernabeu, although Los Blancos remain short of Roma's valuation of the player.