Brazil's number one for the 2018 World Cup enjoyed a fine season in Serie A and has been tipped for a move this summer, with a number of clubs interested in securing his services.
But Liverpool are expected to look at alternative targets as they are unwilling to spent £79million - Alisson's buy-out clause - to strike a deal.
A switch to Anfield has generated the most transfer gossip, with a move to La Liga giants Real Madrid having also been mooted. The Mirror's David Maddock reports on Monday evening that Liverpool "will step up their efforts to land Shaqiri when he returns from the World Cup, after receiving initial encouragement".More news: Photos of Prince George playing with toy gun cause controversy
In-demand Brazilian goalkeeper Alisson hopes his club future will be sorted out before the World Cup starts later this week.
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".
"To be honest, I left the responsibility for all the negotiations to my agent".
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". If it's not resolved before the first game against Switzerland, then it'll have to wait until after the World Cup.