Calling it an "absolute disgrace", Bent added: "I won't stand for that."
A Sunderland spokeswoman told BBC Sport: "Sunderland are liaising with Northumbria Police to fully investigate this matter."
Greater Manchester Police (GMP) appealed for witnesses to come forward as they also launched an investigation into the report of racial abuse.
The Sunderland spokeswoman added: "Behaviour of this kind is totally unacceptable to our football club."
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 Bent's 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.
GMP asked witnesses to the alleged incident to call 0161 872 5050.