Bullard is likely to be officially unveiled on Saturday and paraded to supporters before the FA Cup fourth-round clash against Millwall at the KC Stadium.
The Tigers made their move for Bullard after Bolton pulled out of a potential deal last week and are thought to have beaten off a number of clubs for his signature.
Manager Phil Brown said: "It is a sign of how serious [chairman] Paul Duffen is, how serious Phil Brown is and how serious Hull City is with regards to this Premier League campaign.
"The only way to sell Hull City's plight to Jimmy Bullard is to show him the ambition of the football club, the ambition of the manager and the chairman and play the type of football he likes to play.
"He has been impressed with the way we have approached the Premier League this year and he heard what he wanted to hear."