If the Pope accepts, it would be the second papal visit to the island, Pope Paul VI having stopped here briefly in 1968.
Last September, Argentinean-born Pope Francis made headlines when he prayed for Bermuda’s citizens as his plane flew over the island while en route to Cuba.
Dunkley, on holiday in Italy with his family, delivered the invitation in person on Wednesday when he attended a general audience with the pontiff at the Vatican, pointing out that Bermuda had a large Catholic population.
During his audience, which was arranged by Wesley Spiewak, head of the Roman Catholic Church of Bermuda, the Premier also presented the Pope with a cedar plaque carved by Dwayne Signor, an inmate at Westgate, Bermuda’s maximum-security prison.
Dunkley handed over the plaque, which he said was “beautifully crafted” by Signor.
“I explained who made it, and he seemed very touched,” the Premier told the Royal Gazette newspaper .
The Premier added: “I thanked him for his continued blessings on the people of the world, but also his blessings for Bermuda. He said, ‘And don’t forget to bless me in your prayers as well’.
“I was struck by how genuine and humble he was. He keeps the message simple, he truly cares and I think he gets a lot of enjoyment and satisfaction from what he does.”
Explaining his choice of gift, the Premier said he was “well aware of the talent of many incarcerated inmates” after serving as Minister of National Security.
“Obviously the inmates have made some poor decisions, but I’ve always tried to allow them the opportunity to grow,” he said.
Signor visited the Premier’s office last week to present his plaque. Dunkley told him that it “took my breath away”.
“I said that the Pope would be truly touched, and he was when he saw the artistry and the craftwork,” he added.
Signor was jailed for 10 years in 2010 for shooting a teenager in the back, at a nightclub and seriously injuring him.