The partner, a former Air Force intelligence officer called Summer Worden, has accused Ms McClain of identity theft and improperly accessing her finances, according to the account.
Ms McClain in turn has denied any wrongdoing, arguing that she was simply overseeing the couple's intertwined money arrangements as she had done in the past.
Nasa's Office of Inspector General is now investigating.
The case appears to be unprecedented, with Nasa officials telling the paper that they were unaware of any previous crimes being committed on the space station.
Mark Sundahl, director of the Global Space Law Centre at Cleveland State University, went a step further, saying that he was not aware of any allegations of crime being committed anywhere in space before.
"Just because it's in space doesn't mean it's not subject to law," Sundahl told The New York Times.
He added: "The more we go out there and spend time out there, all the things we do here are going to happen in space."
The couple were married in 2014. Ms Worden had a son who had been born the year before she met Ms McClain.
Ms McClain wanted to adopt the child, according to the New York Times's description of the divorce battle, but Ms Worden resisted.
Rusty Hardin, Ms McClain's lawyer, said "she strenuously denies that she did anything improper" regarding accessing the bank account while in space and "is totally co-operating" with the investigation.
Now back on Earth, Ms McClain has reportedly sat for an interview with the inspector general last week under oath.
Ms McClain, a decorated pilot, was a West Point graduate who flew more than 800 combat hours in Operation Iraqi Freedom before joining Nasa in 2013.
She was due to be part of Nasa's first all-female spacewalk during her time on the ISS, but did not participate in the end.
The case has thrown a spotlight on the little-understood world of space law and what happens if a crime is committed in orbit.
There are rules for what laws govern on the ISS, which has astronauts from America, Canada, Japan and Russia as well as several European nations.
National law applies to each person and their possessions, so an American citizen is subject to American law, while a Russian is subject to Russian law.
The significance of legal rules in the cosmos are only likely to increase as commercial space flights become a reality in the near future.