By Kylie MacLellan

LONDON (Reuters) - Strong will exists in both Britain and Ireland to preserve free movement between the countries following Britain's vote to leave the European Union, Prime Minister Theresa May said on Tuesday after meeting her Irish counterpart.

May, who visited the province of Northern Ireland on Monday, has said she does not want to see a return to past border controls with the Irish Republic, which will be Britain's only land frontier with the EU.

"We benefited from a common travel area between the UK and the Republic of Ireland for many years before either country was a member of the EU," May said at a joint news conference with Irish Prime Minister Enda Kenny in London.

"There is a strong will on both sides to preserve it and so we must now focus on securing a deal that is in the interests of both of us."

The pair said they had agreed that the peace process in Northern Ireland, which ended three decades of fighting between Catholic nationalists seeking a united Ireland and Protestant unionists who wanted to keep the province British, must not be undermined by last month's referendum result.

Northern Ireland voted to stay in the EU, with 56 percent voting 'Remain', putting it at odds with the United Kingdom's overall 52-48 percent result in favor of leaving.

Some have demanded a referendum to split Northern Ireland from the United Kingdom so it can remain in the EU as part of a united Ireland.

Asked about the prospect of a united Ireland, Kenny said he respected Britain's decision to leave the EU and the job now was to work through the Brexit process in "as practical, as imaginative and as creative a manner as is possible".

Kenny also said while many obstacles lay ahead, it was in Ireland's interests to see its neighbor prosper outside the EU.

"We want the upcoming negotiation process to end with a prosperous and outward-looking UK which retains a close relationship with the EU. This is in all of our interests," he added.

(Additional reporting by Kate Holton; editing by Stephen Addison)