The 35-year-old left-sided midfielder was released by fellow Championship side West Bromwich Albion at the end of the 2019-20 season.
Right-back Steven Sessegnon has also joined the Robins on a season-long loan from Premier League club Fulham.
The arrivals of Brunt and Sessegnon take City boss Dean Holden's tally of summer signings to five.
"Chris is going to be very useful to us on and off the field," said Holden. "He's technically very good, with an accurate pass, and is a leader with a strong voice who will be a great teacher to the younger players."
On England Under-21 international Sessegnon, 20, Holden said: "We have been tracking Steven for a while and we are excited about what he will bring to the squad. He will fit into the way we want to play and give us an attacking option down the right side."