The 23-year-old said that he is enjoying being a regular starter under Steve Bruce at the KC Stadium, and feels that he needs to find a club to call home at this stage on his career.
"Of course, there's no reason why I wouldn't look at [joining Hull permanently]," talkSPORT quotes the midfielder as saying. "They've got all the ground work to be a massive club and to go forward.
"It feels as if I need a home now in football. I'm happy coming in each day and doing my business so we'll wait and see in January or at the end of the season.
"I'm prepared for anything in football. I'm enjoying my time here, I love the lads, the manager and the fans. Everyone here is brilliant and I'm getting a run of games so it's so far, so good."
Livermore made his one and only appearance for England in last year's 2-1 win over Italy.