The European Commission will assess the candidates by September but the decision rests with EU leaders who will try to reach a consensus deal at their next summit, in October.
Amsterdam, Athens, Barcelona, Bonn, Bratislava, Brussels, Bucharest, Copenhagen, Dublin, Helsinki, Lille (France), Milan, Porto (Portugal), Sofia (Bulgaria), Stockholm, Malta, Vienna, Warsaw and Zagreb (Croatia) are vying for the medicines agency.
The Council of the European Union said Tuesday that it has received requests from 23 European cities to host two agencies now based in Britain - the European Medicines Agency and the European Banking Authority.
The European Medicines Agency and European Banking Authority are now both based in London.
Six countries - Cyprus, Estonia, Hungary, Latvia, Lithuania, and Slovenia - have not presented a candidate.
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Each of the 27 member states have the same number of votes.
The detail of each tender is available on the website of the Council of the European Union, a body which brings together 28 member countries, sometimes with videos touting the merits of the candidates.
The Council is now going through the submissions and a vote will be taken in November to decide where the two bodies, currently based in London's Canary Wharf, will be moved to.
The Irish application was officially submitted on Monday ahead of a midnight deadline. The country aims to host the European Banking Authority (EBA).
It will also look at whether a country already hosts other European Union agencies to ensure they are not monopolized.
Luxembourg's main competitors are Paris, where the European Securities and Markets Authority (Esma) is located, and Frankfurt, which is home to the European Central Bank and the European Insurance and Occupational Pensions Authority (Eiopa).