Asked if James picked Austin because Bullock has a house there and is often in the city, attorney John Schilling said, "I can't comment on that," People magazine reported. The couple's marriage fell apart after James admitted to an affair.
James has put his Los Angeles home up for sale.
"He intends to move at the end of the month; he's got a need to be there," Schilling told People. James owns a business called the Austin Speed Shop in the city.
The attorney spoke after a custody hearing in Orange County, Calif. court. James and ex-wife Janine Lindemulder, an adult film star, are fighting over custody of their daughter Sunny.
James has custody, but Lindemulder, in Costa Mesa, Calif., says a move to Austin would take her daughter too far away for her to exercise her visitation rights.