According to writer Larry Holcombe, it was Gleason's known interest in UFOs that allegedly led President Richard Nixon to share some information with Gleason, including some UFO data and possibly much more.
According to his wife, Beverly McKittrick, on February 19, 1973, President Richard Nixon took Gleason to an air force base, after the two had played golf together, to show Gleason what he was told were the remains of extra terrestrials.
McKittrick says that sometime around midnight, Nixon suddenly showed up at midnight at Gleason's front door in Key Biscayne. Nixon and Gleason drove through the night to Homestead Air Force Base, 35 miles southwest of Miami. After entering the base, Nixon drove to a heavily-guarded building at the far end of the compound where the two men were allowed into the facility.
According to McKittrick, Gleason told her: "There were a number of labs we passed through first before we entered a section where Nixon pointed out what he said was the wreckage from a flying saucer, enclosed in several large cases. Next, we went into an inner chamber and there were six or eight of what looked like glass-topped Coke freezers. Inside them were the mangled remains of what I took to be children."
McKittrick described the incident as follows: I’ll never forget the night in 1973 my famous husband came home, slumped white-faced in an armchair and spilled out the incredible story to me. He was late. It was around 11:30 p.m. and I’d been worried. As soon as I heard his key turn in the lock of our golf course home in Inverary, Florida. I jumped to my feet and asked, “Where have you been?” His reply stunned me: “I’ve been at Homestead Air Force Base — and I’ve seen the bodies of some aliens from outer space. It’s top secret. Only a few people know. But the President arranged for me to be escorted in there and see them.”Gleason went to to tell her: "And there were the aliens, lying on four separate tables. They were tiny — only about two feet tall — with small bald heads and disproportionately large years. They must have been dead for some time because they’d been embalmed.”
According to Gleason, Nixon told Gleason that these were actually the remains of deceased aliens.
