He told reporters after a 3-0 friendly victory over Austria: "I believe and I hope that everything will be resolved before the start of the World Cup".
It is understood Liverpool are unwilling to meet Roma's £80m valuation of the player - potentially a world-record price for a goalkeeper.
Sky Sports News reported in February that Liverpool were trying to sign Alisson, having initially made an enquiry for his services during the January transfer window.
'To be honest, I left everything in the hands of my agents.
This week, the Brazilian star revealed he hoped to have his future plans resolved before the start of the 2018 World Cup.
Real Madrid have also been linked with a move for Thibaut Courtois, however, and are following his situation in west London closely.
He will start for Brazil at this summer's World Cup in Russian Federation, but before that kicks off he wants a deal wrapped up.
But the 25-year-old has not given up on a possible stay at the Italian capital as stated by GOAL."There are possibilities on which we are working, but always together with Roma and with respect to my club", he said."We will see what will happen this week".
Alisson kept 22 clean sheets in his 47 appearances for Roma last season, with the Giallorossi reaching the Champions League semi-final stage.
He is likely to be Brazil's No 1 when they begin their World Cup campaign against Switzerland on Sunday.