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The 26-year-old goalkeeper earned a Premier League winners' medal on Sunday as the Reds finally got their hands on the title following a 1-1 draw with Crystal Palace.
The Republic of Ireland international featured 10 times in the top-flight this season after Alisson sustained a hamstring injury earlier in the campaign.
However, Kelleher has now revealed during an interview with Optus Sport that he definitely plans to try something new next season.
He said, when asked if he's looking to play regular football elsewhere: "Yeah, yeah definitely.
"I've said it before as well, I feel like I'm good enough to be a No.1 and I feel like I'm good enough to play week in, week out and that's what I'm looking to do.
This season I was lucky enough to play a lot of games but it's definitely something I'm looking at."
Liverpool have already secured the signing of Kelleher's replacement.
Start time, route and how to watch
And Mamardashvili recently revealed he plans to stay put on Merseyside next season even if he will remain second-choice to Alisson, who has previously attracted interest from the Saudi Pro League.
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Mamardashvili said: “They are looking at me as the Liverpool goalkeeper for the next ten years.
I will spend next season at Liverpool, I think it will be good and training with Alisson will help me grow and gain experience.”
Meanwhile, Liverpool's Brazilian has revealed he's "tired of partying" as the Reds have spent the past month celebrating their second Premier League triumph which was secured following a 5-1 demolition of Spurs in late April.
Giorgi Mamardashvili will arrive on Merseyside this summer to battle for a starting role(Image: AFP via Getty Images)
Alisson said: "To be honest I am a little bit tired of partying!
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