The Brazilian joined the Reds in 2018 in a £66.8m move from Roma, making him the most expensive goalkeeper in the world at the time.
Alisson, 31, has won the Premier League, Champions League, FA Cup, EFL Cup and Club World Cup during his 261 appearances for the Reds.
Fabinho, Roberto Firmino and Jordan Henderson all left Liverpool for Saudi Arabia in 2023 but Alisson says he wants to remain on Merseyside.
“I want to honour my contract and finish my contract here or make a new one. I am really happy here. My family is happy,” said Alisson, whose contract runs until 2026.
“I never got to the point where I was talking about wages and things like this. It was just interest, but when you hear about the numbers the other players are getting you are a little bit attracted. That is normal.”
Alisson added “now is not the time” for a move to Saudi Arabia.
Liverpool are closing in on the signing of Valencia goalkeeper Giorgi Mamardashvili to provide competition for Alisson.
The Georgia international, 23, is set to move to Anfield in a deal worth a reported £30m.
“I knew about that [Mamardashvili] before it came out on social media and that is a good message for me because the club cares about what I think,” said Alisson.
“They are doing the right thing but, on my side, as long as I have my contract here and as long as I am happy here, the club is happy with me, my family is happy here, [so] I will stay.”
