William was visiting the sandwich shop on the Dorset coast on Thursday as part of his ongoing charity work tackling homelessness.
Gascoigne said he was having a coffee with his manager when he saw hundreds of people gather ahead of the prince’s arrival.
Speaking to Dan Wootton on GB News, the 56-year-old said: “I was just having a coffee with my manager, Katie.
“So as we were walking up, there were hundreds of people and dozens of cameras and I thought, ‘Well, that’s nice of them to turn up for me.’
“So obviously I went in and I just started talking to him and, I don’t know why, I just said, ‘Prince, it’s Gazza.’
“And he said, ‘Is that you?’ And I went, ‘Yeah,’ and then he said he’d been watching out for us (sic).
“What a great guy he is, and I tell you what, he’s good-looking and all.”
William was visiting Pret to discuss its plans to expand The Pret Foundation’s programme to help 500 people experiencing homelessness get jobs at its shops around the country.
The chain made the commitment as part of its work with William’s Homewards initiative – the prince’s ambitious five-year project to end homelessness in six locations, which launched in June.
Bournemouth, Christchurch and Poole form one of the flagship locations where the Homewards programme is taking place, alongside Aberdeen, Sheffield, Newport, Lambeth and Northern Ireland.
After meeting the footballer, William said it had been a “very interesting morning”.
“We even met Gazza, he popped into (Pret),” he said.
I just thought I couldn't resist, I've got to give him a kiss
Sharing more details about his conversation with William and the kiss, Gascoigne later said: “He just said he’d been looking out for us and making sure I was okay.
“So it was nice of him to go out of his way, not just to speak to us but to put his arm around us and all that, you know.
“So, in fairness, I kept it in the family and I gave him a kiss.
“I felt like I could’ve talked to him for ages and it was really nice for him to actually be in Bournemouth.
“And there were so many people there with the opportunity to meet the guy – and he is just a really nice guy, a really nice guy.
“I just thought I couldn’t resist, I’ve got to give him a kiss.”