I'd be more than happy to sell a gun to someone who admitted to me that he couldn't pass a background check - it isn't the transfer of title that's illegal, it's the transfer of possession. A convicted felon can own all the guns he can afford, he just can't be in a position to "exercise dominion and control" over them. So like grylnsmn said, if they want the gun shipped to an FFL, that's fine, it can sit there with that FFL until they can pass the background check, I reckon. If it's an in-state, face-to-face transaction, and they present a governmentally issued identification document to show that they really are a resident of Virginia, and are willing to sign my bill of sale form (which functions as a sworn statement to the effect that they are legally eligible to be in possession of a firearm), that's about as well as I can do.
The fact that someone tells me they don't know whether they can pass a background check, by itself, has nothing to do with whether they're entitled to be in possession of a firearm. People have so little understanding of this confusing area of law that lots of folks really have no idea whether they could get through that wringer. Lots of folks think they can't, and will in fact be denied, even though they should be able to, and there's no legal reason why they can't (people convicted of domestic battery in a Virginia Juvenile and Domestic Relations District Court are generally not persons who are guilty of a "misdemeanor crime of domestic violence", for example, or people who pleaded nolo contendere to a felony prior to a certain date). The belief that one is unable to pass a background check, by itself, is not a barrier to that person's acquisition of a firearm.
If, on the other hand, he tells you he just got out of Western State Hospital where he'd been undergoing treatment pursuant to a court order finding that he was not guilty of murder by reason of insanity, well, that's a different story.
A guy came to me not too long ago wanting to buy a silenced .22 semiauto pistol, had already placed the order and wanted to pick it up, but the sheriff's office had turned down his request for NFA certification. He wanted me to go into the circuit court and get his gun rights restored. It turned out that he had an extensive history of both voluntary and involuntary hospitalization for psychological problems, some of which involved "violent and/or suicidal ideations". I didn't take that case. I have been successful in other cases, though, where the client couldn't pass a background check because they'd been subjected to an order requiring their detention for psychological observation. The people who do the checks apparently can't tell the difference between "observation" and "treatment". So these folks were "watched" for the requisite 48 hours and then released. No problem as to firearms possession. Can't pass a background check.