Calling it an "absolute disgrace", Bent added: "I won't stand for that."
Related ArticlesWigan Athletic 1 Sunderland 0Steve Bruce: Chris Kirkland England's No 1 goalkeeperDarren Bent fit after hamstring scarePremier League tablePremier League actionPremier League fixtures"Sunderland are liaising with Northumbria Police to fully investigate this matter," a Sunderland spokeswoman said.
"Behaviour of this kind is totally unacceptable to our football club," the spokeswoman added.
Bent, who has scored nine Premier League goals this season, joined Sunderland from Tottenham in August for a club-record fee that could rise to £16.5m.
Before his move was completed the 25-year-old Bent criticised Spurs chairman Daniel Levy on the social networking site over the delay of his transfer.
One of the tweets said: "Do I wanna go to Hull? No. Do I wanna go to Stoke? No. Do I wanna go to Sunderland? Yes."
Bent is an avid user of Twitter and has over 20,000 followers on the micro-blogging site.