Ms Foster said she discussed the issue with Mr Johnson when he visited a vaccination centre in Co Fermanagh earlier this month, and pledged to continue that conversation.
“I think he does understand the difficulties, particularly around border areas and the movement of people in relation to vaccination,” she said.
“If there is surplus vaccine then we should share it with our nearest neighbours out of neighbourliness but also out of the fact it will have an impact of course on us here in Northern Ireland, so there’s a very practical reason why I believe that should happen.”
Deputy First Minister Michelle O’Neill said she would like to see more co-operation.
“What happens across these two islands has implications because we move freely across the two islands, so it is crucially important that we have an all-island approach to dealing with vaccination rollout,” she said.
“It’s the right thing to do, it’s the good thing to do. It’s responding to a global pandemic, and we are all in this together, so we need to work together in order to protect our people. I would like to see a lot more co-operation as we come out of this period and into the future.”
Mr Swann said offering vaccine surplus to the Republic was the “right thing to do”.
“My responsibility is to the people of Northern Ireland so, as of our operation here today, my key aim is to get as many people in Northern Ireland vaccinated as quickly as possible,” he said.
“We hope to have everyone receive their first vaccination by the end of July.
“In the Republic of Ireland, I’ve heard the Taoiseach say they hope to catch up in a couple of weeks or a month’s time, so that’s something we would like to see.”
There were reports in a Sunday newspaper that UK ministers were working up plans to share 3.7 million doses with the Republic.
Speaking at the weekend, Minister for Foreign Affairs Simon Coveney said: “There isn’t an offer that I’m aware of, or that the Government’s aware of, from the UK”.