Intense negotiations will continue over the weekend, with politicians set to receive a third draft of proposals on a way forward by Sunday afternoon.
That will go to the table at 9am on Monday.
"I would simply say that much progress has been made, but much remains to be resolved," Dr Haass said at a press conference in Belfast on Friday evening.
However, it seems that flags in particular are proving to be a real sticking point and it is possible that no clear resolution on that subject will be made after all.
Dr Haass gave himself the end-of-year deadline and, if a deal is not agreed on Monday, his team will return to Northern Ireland on 27 December to strive to find resolutions before the New Year.
UTV Political Editor Ken Reid said that, while there had not been a breakthrough as such, those involved in the negotiations had found there was certainly "enough to keep going".
Earlier, following a round of negotiations on Friday morning, the two main unionist parties voiced their concerns over the proposals currently on the table.
Speaking outside the talks venue at the Stormont Hotel, the DUP's Jeffrey Donaldson said: "Realistically, I don't see an agreement being reached today, I think there is still considerable work to be done across all three areas.
"But we are determined and committed to continue the process until we get a resolution."
Ulster Unionist leader Mike Nesbitt added: "Clearly we are at the point we either say goodbye to a deadline, or you say goodbye to the opportunity.
"And we do not want to say goodbye to the opportunity, because it is still there.
"There's a lot we don't like about this document. We think it's time to park flags because we are not going to agree on that in this round.
"But there is still the opportunity in parades and dealing with the past."
However, Alliance, the SDLP, and Sinn Féin said that they were hopeful of an agreement being reached.
Sinn Féin has said it is prepared to work right up to Christmas Eve, if necessary.
MLA Gerry Kelly added: "The idea was that this would be finished before Christmas and we are still very much up for that.
"We have made advances and we can make progress, but the political will has to be there."
Alex Attwood of the SDLP added: "We have said to Richard Haass and Meghan O'Sullivan that they need to press on with their work and press the parties to move to an agreement."
And Alliance MP Naomi Long said it was important that momentum was not lost.
She added: "There is real scope for a deal to be done on these issues.
"Now is not the time to put the brakes on this process, it is the time to drive it forward and get this deal over the line."
It is thought that Richard Haass' first draft contained a proposal to have the Irish flag fly from Stormont for special visits from Irish delegates.
However, it is understood that any reference to flying the Irish Tricolour has been ruled out in the second draft of the proposals.
A new version was emailed to parties by the talks chairman on Thursday, however, Jeffrey Donaldson described the latest draft as still containing "a few silly things".
Political commentator Brian Rowan told UTV: "In the draft proposal on Monday, controversially Dr Haass had identified occasions when he thought it would be appropriate to fly the Tricolour at Parliament Buildings - a small number of stipulated Irish visits.
"I'm told there is no reference to the Tricolour in this latest draft. There is no advance on designated days for the Union flag in terms of Belfast."
Brian Rowan outlined some of the other details of the latest draft currently under discussion.
He said there have been "some changes" on the topic of parades.
"The 'political and cultural expression and facilitation office' becomes the body where you make applications for a parade," he explained.
"The Department of Justice, the Department of Culture and the Office of the First Minister and deputy First Minister will all have a say in appointing a director for that office."
Meanwhile, he said the issue of the past is a "work in progress".
Mr Rowan explained: "Much of what was in Monday's draft document has found its way into the second draft. Immunity survives with this draft. That fits into the second tier, the independent commission for information retrieval. There will be a use of limited immunity within that process."