The peace building project is set to be confirmed in the coming weeks as part of a £300m redevelopment of the 350-acre site near Lisburn.
Unionist critics say the planned centre will be a shrine to IRA prisoners.
But supporters argue that the project will create thousands of jobs.
The application for the European Peace Three money was submitted in January 2011 and confirmed - with conditions - in December.
A firm financial offer reached the office of the first and deputy first minister last week.
The money comes from the same European fund that provided £13m for the Peace Bridge over the River Foyle in Londonderry.
Since the Maze closed 12 years ago there has been a constant debate about how the extensive site should be used.
It was ear-marked for a new national sports stadium but the proposal was rejected after years of disagreement.
The conflict resolution centre will provide a place for visitors from around the world to exchange views on conflict transformation, a focus for education and research about the troubles together with exhibition space and an archive.
It is envisaged there will be input from ex-prisoners, prison officers and victims.
The centre will sit alongside a preserved H block and other buildings, including the chapel and the hospital where the hunger strikers died.
Unionists have been critical of the preserving of certain parts because of their significance to the republican movement.
Ten members of the IRA and INLA starved themselves to death in the Maze in 1981 and it was also the scene of their 'dirty protest'. Thirty-eight IRA prisoners also took part in the largest prison escape in British history in 1983.
The project's supporters portray it as an important building block in redeveloping an area twice the size of Belfast's Titanic Quarter - an investment which it is argued could generate as many as 5,000 jobs over the next 20 years.
No official announcement on the conflict transformation centre is expected until progress can also be confirmed on the Maze's wider economic regeneration.
That will include a planned move by the Royal Ulster Agricultural Society from Balmoral in south Belfast to a new Maze centre of rural excellence.
It's thought the first agricultural show could take place on the site of the former jail as early as next year.
European funding of £18m has been approved for building the contentious conflict resolution centre on the site of the former Maze prison.