The Reds shelled out a world-record fee for a goalkeeper in capturing the Brazil international from Roma for £65 million earlier in July.
He has now joined Jurgen Klopp's men at their training base in France following a post-World Cup break.
Alisson said on liverpoolfc.com: "It has been a great day which I had been looking forward to. I've had a good few resting days with my family but we all couldn't wait to get the new season under way soon.It's only my first day and I feel at home already."
Alisson's Brazil team-mate Roberto Firmino also reported back for training, and the pair could play a part in Liverpool's two remaining pre-season matches, against Napoli on Saturday and Torino next Tuesday.
The Reds then begin their Premier League campaign against West Ham on August 12, and Alisson said: "Although these two weeks don't seem like a long time, it's enough for us to be ready for our first game against West Ham.
"We'll play two friendlies before this game and I'm sure the hard work the squad have put in so far is going to help me become a member of the squad as quickly as possible so that I am 100 per cent ready for our first game to kick off the Premier League in a good way.
"My first impression tells me it won't be difficult for me here - it's a great squad formed by great players and brilliant professionals. I'm sure it's going to be a great season for us all."