I bought a car with outstanding finance before, it's slightly more hassle than a normal deal but there's no real risk as long as you check it's all legit and deal with the finance company yourself.

IIRC all I needed was the car reg, agreement number and the owners name and address because the seller had already phoned to say I'd be calling to discuss payment... so phoned the finance company, explained that I wanted to clear outstanding balance on the Saturday, they gave me a settlement figure, I told my bank that I was planning to make the transaction that weekend (discovered that was irrelevant depending on account) and on tge Saturday I paid the small extra amount over the finance settlement figure in cash to the seller on collection/completion of docs.

If the owner is adamant that they want to do it after the deal is done then I'd walk away from the deal without a doubt.


Cheers,
Grant