Talks began in September, aimed at tackling the dispute over welfare reform, paramilitarism, and budget matters.
British prime minister David Cameron recently met Stormont’s political leaders as the pace of negotiations stepped up.
The Taoiseach told the BBC: “I am very hopeful and happy that the reports I am getting are that a deal is on here.
“I do hope it can be concluded successfully in the next couple of days.”
A vexed budget wrangle has left the power-sharing administration in Belfast facing an unsustainable black hole of hundreds of millions of pounds.
A resolution to the long-standing impasse over the executive’s failure to implement the government’s welfare reforms in the North will be crucial to any breakthrough.
It is understood that Stormont’s leaders want the British government to commit extra funding to the power-sharing executive, both resource and capital, as part of any settlement.
The wider negotiations, which have been on-going for weeks, are also trying to find a way forward on other problems causing the current instability at Stormont, including the fallout from a recent murder linked to the IRA and a row over how to deal with the legacy of the Troubles.
Mr Kenny is meeting David Cameron at 10 Downing St this afternoon for bilateral talks.
Yesterday, Mr Kenny laid a wreath at the war memorial in Enniskillen, 28 years to the day after the IRA bombed the annual Remembrance Day service.
Eleven people, who had gathered to pay their respects, were killed and dozens more were injured in the no-warning blast in 1987, just minutes before the event had been due to start.
In terrible weather conditions, Mr Kenny joined other dignitaries in laying a wreath at the foot of the memorial.
In Belfast, Foreign Affairs Minister Charlie Flanagan laid a laurel wreath at the Cenotaph.
He said: “I am pleased to represent the Irish Government for the second year at the Remembrance Sunday commemoration at Belfast City Hall.”
More than 200,000 Irish-born soldiers served in the British Army and Navy from 1914 to 1918.