The youngster has been on loan at the Riverside club for the first half of this season.
While his deal is through to the end of the season, Doak can be recalled by the Anfield side.
Asked by Sky Sports if he wanted to stay at the team managed by Manchester United legend Michael Carrick, Doak stated: “Yeah, without a doubt.
“I think we play some of the best football in the league and I think a lot of that comes down the right-hand side.
“I think it suits me really well and I’ve got a great group of lads around me - a lot of talented footballers."
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