from www.tmz.com – Jesse James’ ex-wife, Janine Lindemulder, threatened to kill him … this according to what Jesse’s lawyer just told the judge in their custody war.

Jesse has been testifying on the stand this morning, but hasn’t mentioned the allegation … so far.

As we previously reported, Jesse bought a house in Austin and wants to take Sunny there so she can be closer to Sandra Bullock … who developed a close relationship with Sunny before her divorce from Jesse.

But Janine filed documents trying to stop the move — saying the Sandra excuse is BS because if she really cared about Jesse’s kids … she would have stayed in California.