I don't know of any lender offering assumable loans. There are probably a few owners out there with assumable loans, but they'd be so old that they'd be nearly paid off.
There are a whole lot of ways, though, to simulate an assumable loan. Here are two:
(1) Use a land trust. The mechanics in brief: The owner of a property (let's call him Jim Smith) establishes a land trust and transfers ownership of the property to the trustee. At this point, Jim is the only beneficiary of the land trust. I'm not a lawyer, so this isn't legal advice. However: The transfer from Jim Smith to Jim Smith's land trust does not violate the Garn St. Germain Act or a lender's due on sale clause. Then the would-be buyer is added to the land trust as a beneficiary. If he's living in the property, he's the resident beneficiary. The resident beneficiary is responsible for payment of the mortgage, HOA, etc., to the owner of the property--the trustee. The trustee, in turn, sends payments to the lender, HOA, etc. The resident beneficiary is also responsible for all maintenance and repairs on the property. In return, the resident beneficiary receives the tax benefits of ownership (deduction of mortgage interest, taxes, etc.). This arrangement can continue for as long as the parties wish. At some point, the property is brought out of the land trust and the resident beneficiary buys it. Or, if he chose, the resident beneficiary could sell his beneficial interest to another buyer.
(2) Take the property "subject to" the existing financing. As in the example above, Jim Smith wants to sell his house. Bob Jones wants to buy. Bob asks Jim to transfer the deed of the property from Jim to Bob. However, Bob asks that Jim remain on the mortgage. Bob promises Jim that he, Bob, will pay the mortgage. Bob pays Jim something--maybe $10...maybe $10,000. Bob also pays for any back payments or arrearages of Jim. Now: This can be very dangerous for Jim. He's given up ownership of the property, but is still responsible for the mortgage. Bob's promised to pay it, but if he doesn't, the lender will come after Jim. Note, too, that this does violate the lender's due on sale clause. However, from Bob's perspective, he now is responsible for the same mortgage that Jim had--he's assumed responsibility for payment of the mortgage. And Bob owns the property. At some point in the future, Bob will either refinance the property, or sell it. The "subject to" is a technique many real estate investors use. Done properly, it can be a win-win situation for both parties. However, there are also scams out there in which people acquire ownership, then don't make payments, and the seller and the lender are both harmed. The land trust, on the other hand, provides very good protection for all parties.
Anyhow, there are two ways to simulate an assumable loan.