The transferring of a mortgage from one party to the other is often called an assumable mortgage and is a possibility for someone looking to get rid of a home or property they no longer want. The ability to assume someone else's mortgage is rare. It is usually only attractive to a buyer when the rate the seller has on their mortgage is better than any rate a buyer could get in the current market or if the lender is not requiring a credit check. Closing costs are also generally lower because there are no commissions to pay.
Find out if the mortgage you want to transfer is transferable. Most banks now include "due-on-sale" clauses in their notes, meaning that the balance of the mortgage is due when the home is sold. This prevents buyers from getting the lower interest rate of the seller's mortgage. This keeps the lender from losing out on a possible higher interest rate. FHA and VA loans do not have this clause.
Qualify the assumer for the loan. The bank or lender will want to be sure the person assuming the mortgage can make the payments.
Pay a down payment to the seller and the assumption fee to the lender. The buyer will need to pay the seller in cash for any equity they already have in the home. There also will likely be a fee from the lender for transferring the mortgage.
Make sure the seller is completely removed from all loan documents; otherwise, if the buyer defaults, the lender could go after the original mortgage holder.
Consult an accountant or tax attorney regarding the tax implications of the transaction for all parties involved.