Sinn Fein and the DUP have held "intensified" dialogue on restoring political powersharing in Northern Ireland.
Engagement has been stepped up over the last week between the former coalition partners in devolved government, both parties acknowledged.
Northern Ireland Secretary James Brokenshire said their private discussions demonstrated a clear willingness to speak openly and frankly about issues that needed to be resolved and to explore the potential for finding agreement.
He added: "I was encouraged during my meetings with parties this week that there is agreement on the need for an Executive to be formed as soon as possible.
"They must continue to work together to find a resolution to their differences and secure the re-establishment of devolved government in the interests of everyone in Northern Ireland."
Formal talks between the two largest parties in Northern Ireland have yet to resume following a break for summer.
ble and are therefore ready to re-engage in formal negotiations together, and with the other parties and both governments, to try and reach agreement in a short, sharp and focused negotiation.
"This process should begin immediately."
DUP leader Arlene Foster said engagement with Sinn Fein has stepped up over the last week and there has been detailed contact over a number of days to assess if a deal is possible.
"We intend to continue with a further series of bilaterals with all of the other parties to determine whether agreement can be reached in the short time available," she said.
Mrs O'Neill said it is "absurd and illogical" to accuse her party of pursuing a chaos strategy over the political deadlock.
She said progress is possible on the Assembly stalemate and her party is ready to restart formal negotiations with the DUP, other parties and the two governments.