He played 209 games for Spurs, scoring 15 goals, after joining the north London club in July 2005 from Derby County where he made his debut aged 16 in 2003.
Huddlestone, who has four England caps, has been hit by ankle trouble in recent seasons, undergoing surgery.
Livermore, 23, has struggled to earn a first-team place at Spurs and made only 11 league appearances last season. His one England cap came as a substitute against Italy in 2012.
Hull said on their website (www.hullcityafc.net) that the pair will join the squad for training on Thursday ahead of the club's opening Premier League game at Chelsea on Sunday. ($1 = 0.6436 British pounds)
(Reporting by Tony Goodson; Editing by Ken Ferris)
