The Taoiseach has held talks in Spain with its Prime Minister Mariano Rajoy, during which they discussed the implications of Brexit ahead of the opening of negotiations between the European Union and the UK.
Spain is the first European capital Mr Kenny has visited this year ahead of the opening of the negotiations between the EU and the UK.
Mr Kenny said he outlined Ireland's special circumstances in relation to Brexit regarding trade, the common travel area, the peace process and Northern Ireland.
The Taoiseach also invited King Rey Felipe VI of Spain and Queen Letizia to visit Ireland during his trip.
Speaking at a joint press conference, the leaders said Ireland and Spain will work to defend the rights of their citizens in the wake of Britain's exit from the EU.
When asked if Ireland should be considered a special case in the Brexit negotiations, Mr Rajoy said the Taoiseach had given a very clear reply on the matter and they were faced with a territory which is undergoing a peace process and needs support, and he said they would continue helping in this area.
He described bilateral relations with Ireland as "excellent".
Taoiseach Enda Kenny says Ireland wants a Brexit outcome as close to the status quo 'as possible' after meeting Spanish PM Rajoy in Madrid. pic.twitter.com/hLB6Oeo4Xl— RTÉ News (@rtenews) January 12, 2017
Afterwards a Government spokesman said Mr Kenny had sought and secured Mr Rajoy's recognition of the unique circumstances and issues for Ireland including Northern Ireland.
Mindful of the independence movement in Catalonia, Mr Rajoy has previously ruled out any special deal for Scotland, maintaining the position that if the UK leaves Europe, Scotland must also leave.
Mr Rahoy today strongly ruled out any referendum on Catalonia. He said there is not going to be a referendum which can break up the unity of Spain.
He has also proposed joint sovereignty over Gibraltar after Brexit.
The Taoiseach and the Spanish leader are both members of the European People's Party in Europe and Mr Kenny will be hoping those political ties will shore up Spanish support for Ireland's position.
Jobs, tourism, migration and counter terrorism were also to be on the agenda at today's meeting.
Meanwhile, representatives of the Border Communities Against Brexit group have expressed concern over the possible introduction of a hard border following the triggering of Article 50 by the British Prime Minister Teresa May.
The group's leader, Declan Fearon, noted the responsibility of the Government to bring the views of people living along the border to decision makers in Europe.
Mr Fearon told the Oireachtas Good Friday Committee this afternoon that the island of Ireland should be treated as one economic entity.
He said people living on the border do not want to see the situation in Eastern Europe - where borders are organised by Frontex.
Mr Fearon said people from all sides of the community in Northern Ireland needed to face up to the fact that the a land border is not feasible.
Instead, he said the border needed to be controlled and organised at ports and airports.
Bernard Boyle, a representative from the group, told the committee that there are number of people "rubbing their hands with glee" with the idea of a hard border between the North and South following Brexit.
He also said that smugglers in particular would be pleased with the return of a border. Mr Boyle said: "Those people anecdotally voted for Brexit because it was in their interests, not ours."