Speaking in Donegal, at the MacGill Summer School, Mr Ahern also suggested that Britain after Brexit would undermine its own market as a destination for Irish agri-food exports.
His remarks come as negotiations between the EU and Britain resume this week on attempts to agree a withdrawal deal and therefor avoid a cliff-edge scenario for Brexit.
Mr Ahern, delivering the 18th Annual John Hume lecture in Glenties, said: “I would strongly urge the European Union to continue to support future peace programmes in Ireland, post the British departure from the European Union.”
Discussing Brexit, the former Fianna Fáil leader said there was “no reason why the European Union cannot continue to support the peace process in Ireland within the changing new political architecture of the European Union.”
EU funding has helped government agencies in the North as well as public bodies, private sectors companies and the farming community there.
Under the Common Agricultural Policy 2014 — 2020, farmers in the North will receive in excess of €2.6bn in direct payments during this period. This accounts for 87% of total farm income in the North.
However, billions of euro in funding will come to a head at the end of 2020, the audience was told.
“The agriculture and food sectors are vitally important components of the economy of Northern Ireland and these sectors will need to be heavily supported in financial terms post the British departure from the European Union,” said Mr Ahern.
Brexit had taken away the “debate about the replacement of the future financing requirements of existing EU funded programmes,” Mr Ahern stressed.
He then added: “In other words, how much money will the British government allocate post 2020 to substitute EU financed programmes that operate in Northern Ireland at this time?
“This is a very serious issue indeed and all contracting parties need to sit down and assess how to move these matters forward in a positive and constructive manner.”
Speaking to the media ahead of the lecture, Mr Ahern said we were only a “matter of months away” from where negotiations on the next round of funds for the North were to start.
“They [Britain] have said the will provide. There is a big difference between that [and reality]...It could be left where it is not paid.”
There were also warnings about the impact of Brexit on Irish food exports. Mr Ahern added in his lecture at the summer school: “The shared land border between the Republic of Ireland and Northern Ireland has resulted in the development of a highly integrated agri-food sector, with large volumes of trade annually in live animals, finished products and products requiring further processing.
“For example, over 800m litres of milk is sourced from farms in Northern Ireland, but processed into dairy products and ingredients by co-operatives located south of the border.
“Britain, by conducting its own bilateral trade deals with third countries, will undermine the British market as a destination for Irish agri-food exports.”
Meanwhile, speaking at the opening of the 38th year of the summer school, French ambassador to Ireland Stéphane Crouzat said the backstop, in the opinion of Paris, was the “only viable solution for Ireland”.
This guarantees a frictionless border in the North if it stays in the customs union but it is considered a last resort option.
France was preparing for “all eventualities”, including a no deal, added Mr Crouzat.