Spanish reports claim that Real Madrid have agreed a five-year deal with Alisson.
Madrid are going to have to better their offer by £18 million and they have already agreed personal terms with Allison.
Real have made a £52m bid, with £4.4m of that in bonuses, but Roma are holding out for a £70m transfer fee, Mundo Deportivo
Alisson, who kept 22 clean sheets in 49 outings for Roma in Serie A and the Champions League last season, is under contract at Roma until 2021 and remains a target for Premier League duo Liverpool and Chelsea.
The Goalkeeper was keen to resolve his situation before World Cup.
'I believe and I hope that everything will be resolved before the start of the World Cup,' he said following Brazil's 3-0 friendly win over Austria on Sunday.
'To be honest, I left everything in the hands of my agents.
'If it's not resolved before the World Cup, then it will be discussed after it because my head will be focused only on Brazil.
'There are many possibilities on which we are working, but always together with Roma and with respect to my club. We will see what will happen this week.'