The former Atomic Kitten member is said to have made the request when the two of them met on the set of This Morning last month.
Marsh told Heat: "I said to her, if she's going to do it, she can't mess about.
"She then asked if she could move into my house, as that way I could keep an eye on her. I said yes. It would have to be as strict as that. There's no point training and eating healthily with me, and then going off and eating crap."
The reality star went on to reveal that Katona had visions of competing as a bodybuilder, explaining: "Kerry did say, 'What if you got me so fit I could get on stage at the end and compete?'"
"I could definitely do that with her," she confirmed.
Marsh does, however, feel that Katona has a long road ahead of her, saying: "With Kerry, she probably hasn't got any muscle, so to build it up you need four months to bulk up any muscle then strip away the fat."
Katona recently revealed that she had a new 'tummy tuck' operation in May, in order to reduce stretched skin after her pregnancies.
Marsh has admitted that, despite her new body confidence, she has not had sex for over a year.