The 23-year-old Tottenham midfielder is spending the season at the KC Stadium after a number of high-profile arrivals at Spurs shunted him down the pecking order.
Livermore had already spent loan spells at Derby, Peterborough, Ipswich and Leeds before his first-team breakthrough at White Hart Lane and admits that he is enjoying being a key part of Hull's plans.
Manager Steve Bruce has already indicated he would be interested in a future deal and the player appears to be open to it as well.
"I'm quite prepared for the question to be asked," he said.
"Of course, there's no reason why I wouldn't look at it. They've got all the ground work to be a massive club and to go forward.
"That's what it feels like, 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 has enjoyed his partnership with his former Tottenham team-mate Tom Huddlestone, who signed permanently for Hull in the summer.
The fact that both had to leave the club to achieve their goal of regular playing time is symptomatic of the struggles encountered by many young English players at Champions League-chasing clubs.
New England winger Andros Townsend is making headway in Spurs' star-studded squad but Livermore believes maintaining his place could be the hardest part.
"It's very difficult to stay at a big club. There are great big expectations from the club and the fans and the manager are under pressure," he said.
"It's almost an easy option to bring in a player in who's fairly established rather than take a chance on a younger lad.
"Tottenham have spent an awful lot of money in the past window but there's a lot of talent in the country and at Tottenham and their two breaths of fresh air this season seem to be Andros and Harry Kane, who cost them nothing."