There are only a few ways "around" the Fannie Mae/Freddie Mac buy and bail guidelines which state that the only way to use the rental income to offset the mortgage payment is if you have 1 of the following.
1) A history of receiving the rent, documented by tax returns
2) 20-25% equity in the rental property.
Most lenders have applied these guidelines as overlays to FHA and VA mortgages as well.
FHA may allow the use of a lease agreement if you fall under one of the exceptions. For example, if you are transferred or have to relocate for work.
Conventional mortgages are generally sold to Fannie/Freddie so unless this lender is going to hold and service your loan they still have to underwrite to Fannie/Freddie guidelines.
Who is the lender that is saying they can do this?
John did a good job giving you some clarity. Unless one bank is holding onto your loan, you would need at least 25% equity in the house if you are a first time landlord. Thats a fannie/freddie/fha/ rule. John gave you a good idea of how to get around of it if you can...
When you made the decision to rent your current house and buy a new one to live in, your current home became investment property. How investment property income is treated for purchasing a new home you will occupy varies from lender to lender and loan product to product, whether your new loan will be conventional, and whether the lender you use will sell the loan in the secondary market or portfolio the loan. You need to have a clarifying conversation with each lender. (Loan guidelines can vary from lender to lender.)