Assume you do work out a contract of sale, its probably not acceptable to the ncurrent lender who can then foreclose if they discover it. To be safe, YOU would want something recorded.Sometimes we used to use a 'memorandum of option' that didnt really give much info. Remember the lender may have an acceleration clause. That allows them to take the property if its sold without the loan being fully assumed according to the lenders requirements. If you are paying more than market rent at this time, you are possibly being taken to the cleaners.
As Shanna says, real estate agreements are not enforceable unless they're in writing, and if YOU were to die or something, your heirs would have no claim to the property.
All the best,
Many people have misunderstandings about how improving your financial situation is done. Don't be a victim of the "they say". Get it straight from a mortgage broker. A good mortgage broker will do it for free. If they want a fee for working with you on a plan, walk away.