The force said the cyclist was treated at the scene but his injuries are not believed to be serious.
GMP did not confirm whether the driver had stopped, saying only that inquiries are ongoing, but a witness said the vehicle had left the scene.
The spectator told the PA news agency it was a hit-and-run incident at the intersection between Chorley New Road, Tudor Avenue and Somerset Road.
Footage showed the athlete being treated by an ambulance crew while other participants continued with the running and cycling legs of the event.
The Ironman triathlon involves athletes aged 17 to 70 racing in a 2.4-mile swim, a 112-mile bike ride and a full marathon.
Sunday was Bolton’s 12th time hosting the event and competitors raced in Pennington Flash Country Park for the swim.
Bolton Council’s website said: “For only the second time athletes will cycle into Bolton town centre.
“The bike course will see them cycling along Chorley New Road, passing Queens Park, coming into Bolton town centre and on to Le Mans Crescent, before heading to Bury via Folds Road and Tonge Moor Road.
“The marathon route will take place in and around Bolton town centre and Queens Park, with the finale on Victoria Square, finishing against the backdrop of Bolton town hall.”