The US diplomat, Dr Richard Haass and his colleague, Professor O'Sullivan are expected to return to Northern Ireland at the weekend to make a final attempt to broker an agreement among the political parties.
Last Tuesday, Dr Haass said that consensus had not been achieved in any of the three contentious issues.
The three contentious issues are: flags, parades and how to deal with the past.
Dr Haass has been working on the initiative since September.
He came to Belfast on the invitation of first minister, Peter Robinson and deputy first minister, Martin McGuinness.
Following eight hours of unsuccessful discussions, last Tuesday, Dr Haass said that consensus had not been achieved in any of the three areas.
He is expected to present revised proposals to delegations from five parties, the DUP, the Ulster Unionist party, SInn Féin, the SDLP and Alliance.
The two unionist parties had the greatest amount of difficulties with the range of suggestions put to them in recent weeks.
Dr Haass and Professor O'Sullivan both indicated they have work commitments elsewhere in January and they will not be able to give any further time to the Northern Ireland issues.
It is expected the next round of negotiations will reach a critical point on Monday at which stage a decision will be made on the possibility of a deal.