National Rifle Association Chief Executive Wayne LaPierre was in discussions to have the group’s then-outside advertising agency help him buy a Dallas mansion last year for more than $5 million, but the deal fell through, according to people familiar with the matter.

The aborted house deal and conflicting explanations for why it fell apart are coming to the fore as the New York attorney general’s office is probing the NRA, including Mr. LaPierre’s dealings with the ad agency. The ad agency, Ackerman McQueen, recently turned...