"On one of the trips to America I got alongside IMG and we talked to them about it and over a period of about two or three months and, after meeting Arnold up there, we came to an agreement to invite him to come to New Zealand to play in the New Zealand Airlines Golf Classic at Titirangi in 1978."
Getting the superstar across the line wasn't as hard as you might think.
"He didn't charge us a huge fee," Gleeson says. "He told us he had always wanted to come to New Zealand. We came up with a concept that he would make a film while he was down here. He would play in the tournament and then we would fly him around New Zealand with an IMG filming crew and turn that into a film that we could show in flight and we could turn into a tourism film featuring Arnold Palmer going to all the great places in New Zealand.
"NAC even put a special Fokker Friendship at his disposal with crew and all that. He went to the Bay of Islands, to Rotorua and he played cricket - we put him in whites and he went out and batted.
"So we made this film but we couldn't decide what to call it. Arnold came up with the idea of calling it 'a simpler place and time' and that summed up New Zealand when you think about it.
"We had this little book in full colour and we produced thousands of copies and then distributed it to Rotary Clubs and things like that in the United States."
Palmer enjoyed his time in New Zealand by all accounts and the people certainly enjoyed having him here. Gleeson says he was the most charismatic sporting character he ever dealt with and he almost had an aura about him.
"He was just so friendly with everybody. He played in the pro-am with the executives.
"People just loved him. He was very photogenic, very sociable.
"On the fairways he would play his first shot and then while he was waiting for the others to play their shots he would walk across to the gallery and chat away with them. He had that ability to switch on and switch off.
"We had huge crowds - the biggest crowds we ever had for that tournament."
And he also delivered on the course too.
"He performed so well. He was about four shots behind going into the back nine and he hitched his pants up and he came with a rush to finish third. Bob Charles won that year but Palmer made a charge."