The 33-year-old Hinrich has spent nine of his 11 NBA seasons with the Bulls, including last season when he averaged 11.8 points and 5.2 assists in 32.3 minutes a game.
"Kirk has always been a big part of creating the professional culture we want day in, day out surrounding our club," Bulls GM Gar Forman said in a statement. "He helps our team in so many ways and Kirk had other opportunities elsewhere, but we are very happy he chose to remain in a Bulls uniform."
Bairstow was selected with the 49th overall pick in last month's draft.
"We like Cameron's combination as a player with his size, energy and physicality," said Bulls GM Gar Forman said in a statement. "He is a hard worker who will only get better with time."
It is still unclear how much of a factor Bairstow will have this season, but Bulls executive VP John Paxson said the organization fielded several offers for draft night swaps when Bairstow was selected.
"We were getting calls leading up to that pick with teams offering us a ton of cash," Paxson told ESPN Chicago 1000's "Waddle & Silvy Show" last month. "For people that sit there and say that the Bulls always make decisions based on money -- we never even considered taking a deal like that [on draft night]. And [owner Jerry Reinsdorf] is sitting there in the room -- because we were looking at a player that we think can maybe play a role for us down the road. And we sat there and bypassed [the offers]."
With Bairstow's deal on the books, now the Bulls must officially sign first-round pick Doug McDermott and free agent Aaron Brooks. They also must continue to look for another backup big man to take the pressure of Joakim Noah. Bairstow may eventually fill that role, but that will be determined over time.