The 34-year-old joined Liverpool from Roma in 2018 for £66.8 million and has since made 347 appearances, establishing himself as one of the most important players of the club's recent era.
His current agreement expires at the end of the season, meaning Liverpool face an important decision over the future of a goalkeeper who has helped them win the Premier League, Champions League and several domestic trophies.
Alisson's latest comments came after an impressive performance in Sunday's 3-1 victory over Crystal Palace, when he produced several important saves to help the Reds secure three points at Selhurst Park.
The result was particularly welcome following a difficult week in which Liverpool had suffered defeats against Arsenal and Fenerbahce.
Asked whether he wanted to remain at Anfield beyond the current campaign, Alisson made his preference clear.
“Yes [I want to stay], in my mind. I'm focused on playing every game but in my mind I'm happy to stay.
“That depends on a lot of aspects and different situations. But my will is to stay - I'm happy here and my family is happy here.
“I'm still hungry for titles and to delivering good performances. I still want to play at the top level for as long as I can, so that's my thinking.”
Although the goalkeeper has not personally held discussions with Liverpool about a new contract, he confirmed that his representative is already in contact with the club.
“No, but my agent is talking with the club.”
The situation follows interest from Juventus earlier in the year, with Alisson having considered a move to Italy before ultimately deciding to remain at Liverpool.
The club subsequently activated an option in his contract, extending his stay through to the end of the current season. However, the arrival of Giorgi Mamardashvili has also created an additional consideration for Liverpool as they plan their long-term goalkeeping position.
Head coach Andoni Iraola has previously expressed his hope that Alisson will remain beyond the campaign, although he acknowledged that he could not predict whether this would be the Brazilian's final season at Anfield.
Alisson's immediate priority is helping Liverpool find greater consistency. Despite their quality and ability to produce impressive performances, he believes the team have not yet reached the level required to challenge consistently for major honours.
“Our reality is that we're not doing as well as we can. It doesn't mean that we are not doing well - we are doing well in so many games - but the consistency is not there yet.
“We have a top manager, top players. We are capable of great things but as a team we need to get to a point where we feel: 'We are OK now. We have the confidence. We have everything that's needed to win.'
“We are building that. I hope we get there really quick but it takes time.
“We have to be patient but as players, in our mind, we don't have time to waste.”
Liverpool will therefore have two important objectives over the coming months: improving their performances on the pitch and reaching an agreement with a goalkeeper who clearly wants his future to remain at Anfield.