But there's a problem: An Orange County, Calif., judge has yet to decide in an ongoing custody trial whether to allow Sunny, 6, James's daughter with ex-wife Janine Lindemulder to leave California, where Lindemulder lives.
"Jesse's moving either way, and ideally Sunny is going with him," James's attorney John Schilling told PEOPLE outside the courthouse. "I think Sandra Bullock's role is irrelevant here. Of course she has a lot to offer his kids so it's a fringe benefit to bring them to Texas. But the focus in this case is Jesse and Janine as parents. Janine still has a lot to prove."
James testified that Sunny visited him while he was in rehab in Arizona, and that at one point the child stayed with Bullock in Texas while he was receiving treatment. Sunny has phone conversations with Bullock "two or three times a week" and they sometimes have iChat conversations, he added.
He called Austin "a better place to be," without the prying lenses of paparazzi. "We have a big gated house with lots of land," he said. "Sandy and I spent time there and never once had a problem."
Lindemulder, 41, a former porn actress and recovering drug addict, has accused James of moving to Texas for the sole purpose of winning Bullock back at the expense of Lindemulder's relationship with Sunny.
"Jesse treats me with great disregard as a mother, he doesn't want me in the picture," Lindemulder testified Tuesday. "I believe I'm trying to be replaced by Sandra Bullock. I can't compete with what they have."
Lindemulder has repeatedly stated that she wishes to co-parent with James, Bullock and James's other ex-wife Karla James. Lindemulder testified that Sunny "loves [Bullock] very much," but added, "Jesse and Sandra have not reached out to me."
James has repeatedly raised concerns over Lindemulder's past drug use, but she has denied relapsing and doesn't abuse prescription drugs, she has said.
The judge is considering three options: Allowing James to move with Sunny to Texas, leaving Sunny with Lindemulder in California, or having Lindemulder relocate to Texas along with James.