Alisson has recently returned from the World Cup in Russian Federation where he appeared for Brazil.
The 25-year-old stopper has spent two years in Italy following his arrival from Internacional in his homeland.
For too long Liverpool have come unstuck due to the questionable abilities of their goalkeeper, with uncertainty and uneasiness spreading through the team like wildfire.
The addition of Virgil van Dijk at the centre of defence has solidified one issue area, but the arrival of Alisson would serve to make the Reds' defence more fearsome than their rivals would be willing to give them credit for.
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Mr Mielekamp said the club was optimistic that the trial could be extended. Mr Rallis said: "If he's competitive, he will lift our A-League profile".
Roma are rubbing their hands at the prospect of a bidding war for their prized keeper, who they value at 75 million euros - a price which Real Madrid, who have also been tracking the player, withdrew their interest.
However, Liverpool are now in pole position and the Express are citing a report from Gianluca Di Marzio that claims Alisson has already held talks with Klopp about a move to Anfield this summer. He was named in last season's UEFA Champions League squad of the season.
Fair play for taking it in the spirit it was intended, Liverpool fans.
Alisson, 25, had a breakout campaign for Roma last season where he was considered one of the best goalkeepers in Europe.