On Monday, Mrs May met the first and deputy first ministers on a one-day visit to Northern Ireland.
She stressed that no-one wanted a return to the "borders of the past".
Her meeting with Mr Kenny is understood to have focused on the impact of Brexit on British and Irish relations.
Earlier in July, shortly after her appointment as the new prime minister, Mrs May spoke to Mr Kenny by telephone and agreed to build on the "strong relationship"  between the UK and the Republic of Ireland..
Northern Ireland Secretary of State James Brokenshire described Mrs May's meeting with Arlene Foster and Martin McGuinness on Monday as "positive".
He also said there was agreement about protecting the current border arrangements between Northern Ireland and the Republic of Ireland.
"I think that the prime minister had a really helpful exchange with the first minister and the deputy first minister and I think there is a strong will and strong commitment to not see the return to the borders of the past," he said.
"Indeed that's a theme that underlined discussions that I previously had with the taoiseach on Friday."
Mr Brokenshire added that there were "treaties and various other mechanisms" in place that underlined that there was a "different arrangement" between the United Kingdom and the Republic of Ireland.
Deputy First Minister Martin McGuinness said he hoped there would be an "arrangement which will see the people of the north of Ireland continuing to see their future in Europe as part of an imaginative solution."
The DUP MP Gavin Robinson said there were "two aspects" that had to be considered relating to the border.
"So far, discussions have focussed on the movement of people, there will also need to be a discussion about the movement of goods, customs and whether or not there is a form of a single market or not, whether there is a disparity in tariffs or not and if that is the case, then how you control that," he said.
"So whilst I see no difficulty in getting an arrangement that builds on the Common Travel Area and that allows you and I to move freely into the Republic of Ireland and similarly people from the Republic of Ireland can move freely into the United Kingdom, knowing that we have a uniform immigration policy, it may well be that further discussions will need to take place."
Prime Minister Theresa May has held talks with Taoiseach (Irish Prime Minister) Enda Kenny at Downing Street.