The Republic is one of four countries officially competing to attract both agencies.
The FT reports that Davis also believes that the United Kingdom can also retain another key EU institution, the European Banking Authority (EBA), also located in London.
"There will be an opportunity for the United Kingdom citizens to express themselves on how they see the future relationship between the their country and the European Union", the liberal former Belgian prime minister said on his Facebook page.
Davis is understood to want to include the future of the EMA and EBA as part of the wider negotiations on future trade relations between the United Kingdom and Europe.
The British government's department for exiting the European Union, headed by David Davis, had claimed that the future location of the agencies would be decided during Brexit negotiations. It remains to be seen how these negotiations will ultimately play out. Cities including Frankfurt, Milan, Amsterdam and Paris are thought to be possible new locations for the two bodies. There are many interested member states.
However it is not clear just how quick this decision can be made, though it is unlikely to be confirmed before June. EMA is the European equivalent of the US Food and Drug Administration and has operated from London since 1995.
As of March 29, the United Kingdom government has approved the Brexit bill, and British Prime Minister Theresa May has initiated Article 50.
"This means there will be an opportunity for the United Kingdom citizens to express themselves on how they see the future relationship between their country and the EU".
"This will allow the EU27 to start negotiations", he adds.
European Union officials doubt final decisions can be made in the coming months but say leaders do want to allow time for new infrastructure in other cities before Britain leaves the EU.
EU officials said European Council President Donald Tusk, who chairs summits of the 27 other EU leaders who will decide where to put the agencies, aimed to set out some procedures for making the choices by the time they meet next on April 29.