Media playback is not supported on this device
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."

The boy, 17, shocked passengers on board the Copa Airlines flight from Panama City as he flung the door open before leaping out on Tuesday.
He was confronted by construction staff working nearby and held until police arrived to arrest him.
The boy, a US citizen, was not harmed.
Passengers said the teenager seemed fidgety and anxious throughout the flight and that he had acted very quickly, making it difficult to stop him, according to the Associated Press news agency.
End of Twitter post  by @LisaAminABC7
One eyewitness, Sophia Gibson, told San Jose's KNTV that people were shocked when the passenger leapt through the door and onto the plane's wing.
"It was as if he was like flying out, like it was really fast," Ms Gibson said.
Another passenger, 11-year-old Andrea Sepulveda Guzman, told CBS San Francisco that people had started to scream.
"This guy, I see him jump on the wing and then he starts climbing down the wing and then he just runs. And then he just starts running the other way. And everyone was just screaming," she said.
It was not immediately clear why the teenager had leapt from the plane and run.
San Francisco Airport spokesman Doug Yakel said the boy was travelling alone, adding that no runways or flights were affected by the incident.
A Copa Airlines Flight 208 crew member closed the emergency door and the plane taxied to its gate, where all remaining passengers alighted without incident, the Panama-based airline said in a statement on Tuesday.
The airline added that an investigation was under way and that it was co-operating with the authorities.

The Westgate development in the city centre will eventually have more than 100 shops, 25 restaurants and cafes, a cinema and 61 flats.
The multi-storey car park on the site was demolished last year. It will be replaced with a 1,000-space underground car park.
The development is due for completion in autumn next year.
It will include rooftop terrace dining and new public spaces.
Site owner Westgate Oxford Alliance, a joint venture between commercial property company Land Securities and The Crown Estate, said the development would cover 800,000 sq ft (74,322 sq m) when complete.
Preparatory work began in February 2015 following planning approval by Oxford City Council.
Roy Keane had "no hesitation" in agreeing to remain Republic of Ireland assistant boss once manager Martin O'Neill committed to another two years.
[NEXT_CONCEPT]
A teenager has been detained after opening the emergency door of a passenger plane and sliding down a wing onto the tarmac minutes after touching down in San Francisco, officials say.
[NEXT_CONCEPT]
Work has begun to demolish a 1970s shopping centre in the centre of Oxford as part of a £440m redevelopment.