Following talks with Protestant politician Ian Paisley Monday, Roman Catholic leader Sean Brady said a political deal in Northern Ireland is "within our grasp".
Paisley's party is seen as holding the key to an agreement.
"A lot of progress has been made. Hopes are now rising for further progress. I pray that these hopes may not be dashed but realised abundantly," Archbishop Brady said.
The meeting comes ahead of crucial talks this week on restoring local government to Northern Ireland.
London and Dublin will host negotiations in Scotland in a final push to persuade Paisley's Democratic Unionist Party to share power with the IRA's political ally, Sinn Fein, which is mostly Catholic and opposes British rule in Northern Ireland.
Paisley has so far refused to govern with Sinn Fein, the main party representing Northern Ireland's Catholic population, until he is convinced the IRA, which last year pledged to surrender its arms, has given up violence for good.
Brady, leader of all Catholics on the island of Ireland said the meeting with Paisley had been "helpful and constructive".
Paisley said in a statement the meeting had been a "good and useful exchange of views".
The politician-cum-cleric, leader of the breakaway Free Presbyterian Church he founded in the 1950s, said it was in everybody's interests to "develop the foundations for stability and prosperity".
Political agreement remains elusive, although three decades of violence in Northern Ireland largely ended with the 1998 Good Friday Agreement, which established a short-lived power-sharing executive.
The SDLP, Northern Ireland's second-biggest Catholic nationalist party after Sinn Fein, welcomed Monday's meeting, which it said signalled a new direction from the DUP.
"I hope the meeting is an indicator of a genuine intent within the DUP to go into the St. Andrews negotiations (in Scotland) seeking a positive outcome," said Alasdair McDonnell, an SDLP lawmaker in the British parliament.
