The former Tottenham star joined the Tigers in 2014, following a strong loan spell the previous season, and has been a key man for the club during the current campaign.
Livermore has made 24 appearances in all competitions, and has occasionally deputised in defence due to a host of injuries.
However, recent reports have linked him with a potential move away from the KCOM Stadium, with Crystal Palace thought to be eyeing a deal for the one-cap England international.
Despite the 27-year-old currently having a year left on his contract, Watford are now also thought to have stated an interest in the player.
With Roberto Pereyra and Valon Behrami currently suffering with injuries, Hornets boss Walter Mazzarri is believed to be considering Livermore as an antidote to his midfield problems.
Sunderland have vowed to 'rise again' after their relegation to League One was confirmed on Saturday afternoon.
Diego Simeone has admitted he expects Diego Costa to miss the first leg of Atletico Madrid's Europa League semifinal against Arsenal.