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The 23-year-old moved to Steve Bruce's side in August and has made an impressive start to his spell with the Premier League newcomers.
Livermore is due to return to parent club Tottenham at the end of the season, but he will consider extending his stay at Hull.
"I am really enjoying my time at Hull with the lads, the manager and the fans," Livermore said. "Everything here is brilliant.
"I am open to staying permanently. There is no reason why I wouldn't look at it, so I am prepared to be asked the question."
Livermore has also spent loan spells at six Football League clubs, including Derby County and Leeds United, after struggling to break into Tottenham's first team.
