If this worked, don't you think that a lot of people would be doing it?
The "CATCH 22" in the equation is the MONEY that is missing; the difference of the Market Value of the property, and the amount that is OWED: Someone has to come up with that money to keep the house out of Foreclosure, or, to purchase the house before it goes to foreclosure.
Think about it.
Or if you mean in foreclosure as it has gone through the final foreclosure step, usually the Sheriff's Sale, where the lender has bid what the owners owed against it and now is lender owned or REO? If so, then that lender has investors who want their money back, as much of it as they can get and as fast as they can get it. Renting is not an option to an investor. The lender has spent a lot of money on attorney's and other costs with a foreclosure; unfortunately, renters have no money in a property and just living in a house makes wear on carpets, paint, everything.
The only time lenders agree to rent is if there is a lease in effect at the time of foreclosure (Sheriff Sale) - and they only honor it until it expires.
No. Lenders are not in the business of being landlords and if the current owner cannot make the payments (hence, foreclosure) you would be paying the current owner and the bank would still foreclose if the current owner does not pay the bank.