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O'Neill and Keane agreed new deals with the Football Association of Ireland (FAI) in midweek although contracts have yet to be signed.
The duo's Irish side begin their Euro 2016 campaign against Sweden on Monday.
"We're happy to stay on. I really enjoy the job, working with the players, the staff and the FAI," said Keane, 44.
The former Manchester United star has made no secret of his desire to return to club management at some stage after previously managing Sunderland and Ipswich.
During his Republic of Ireland stint, which began in November 2013, Keane has twice been linked with the Celtic job but opted to remain as O'Neill's assistant on both occasions.
"With the club stuff, I'm 44 years of age so there is plenty of time for that," he said.
"I am keen to get back into it - and when I said short term, I mean in the next few years, not the next few weeks."
In an otherwise good humoured news conference at the Republic's tournament base in Versailles, Keane's brow only furrowed when he was asked whether his contract includes a clause that will allow him to talk to clubs over the next two years.
"I think what's written in my contract is none of your business," the Corkman replied to the journalist.
"Who do you work with? What's your deal? Do they pay you good money? It's none of my business."
A relaxed Keane even played along with the inevitable question about the Republic's tournament facilities, 14 years after he was infamously sent home from the county's pre-tournament training camp in Saipan on the eve of the 2002 World Cup.
"The FAI have done brilliantly. No doubt lessons have been learned. Trying to find something that was wrong, then maybe my bed's a bit too soft," joked the Republic assistant.
"The facilities are brilliant - they're a credit to everyone in the set-up here; the training pitch, the hotel, the location.
"It's great. It really is. We're enjoying it."
Roy Keane had "no hesitation" in agreeing to remain Republic of Ireland assistant boss once manager Martin O'Neill committed to another two years.