Sinn Féin's President Mary Lou McDonald said "significant progress has been made" on the protocol during negotiations between the EU and UK.
She made the comments after meeting with the Prime Minister on Friday.
Rishi Sunak is making a short trip to Northern Ireland to speak to all Stormont parties and provide them with updates on the current negotiations.
Meanwhile SDLP Leader Colum Eastwood said Mr Sunak gave "scant" detail any potential deal.
The Prime Minister's visit comes as there is mounting speculation that a deal between the EU and UK could be unveiled early next week.
Following his visit to Northern Ireland, Mr Sunak is set to join European leaders in Germany this weekend for the Munich Security Conference and the protocol is likely to feature in discussions on the margins.
Meetings between the parties and Mr Sunak are set to last around 15 minutes each.
The DUP removed its first minister from the Northern Ireland Executive in 2022 due to its concerns over the Northern Ireland Protocol. The post-Brexit trade deal resulted in some checks being carried out on goods travelling from GB to NI. The DUP set seven tests which it said any protocol resolution would require in order for it to consider re-entering power-sharing institutions in Northern Ireland.
After meeting with the Prime Minister, Sinn Féin's President Mary McDonald said: "It is clear now that significant progress has been made and we're very heartened by that."
"We now want to see a speedy conclusion of matters and above all else we want to see the institutions restored, government restored, here in the north.
"The bottom line is that we have to ensure that any deal provides for ongoing access to the European Single Market, no hardening of the border on the island of Ireland and a protection of the Good Froday Agreement in all of its parts, those are the core elements and aspects that need to be protected."
Meanwhile SDLP Leader Colum Eastwood said he believed Mr Sunak was "ticking the box" of engaging with the Stormont parties.
"I think he's very careful not to get into too much detail until the deal is done and I suppose that's fair enough," he said.
Mr Eastwood said he made clear to Mr Sunak that the dual market access provided for in the protocol, allowing business in Northern Ireland to sell unfettered into the EU single market, must be preserved.
"He said the deal is not done yet," he added.
"I think he's clear that lots of progress has been made and that's what we've been hearing from the European side and from Dublin as well. But he says it's not done and he's going to Munich to see Ursula von der Leyen (European Commission president) and we'll see what comes out of that.
"But I would be fairly optimistic that we're very close to an agreement.
"We have to be courageous and we have to take steps that allow local governance to be back up and running to deal with the health service and to pick up the opportunities that the protocol provides for the economy."
Alliance Leader Naomi Long said that she explained to the Prime Minister the importance of dual-market access for Northern Ireland.
"We were trying to convey to him the importance to local businesses, and to Northern Ireland as a whole, of having a relationship with the EU that is positive and constructive, of having stability, of maintaining dual access in terms of markets, and on us being able to move forward and restore the Assembly," she told reporters.
Last week the Assembly was recalled in a bid for parties to nominate a speaker and to progress the implementation of opt-out organ donation law in Northern Ireland. The legislation has been named Dáithí's Law after six-year-old Dáithi MacGabhann who is waiting for a heart transplant. The recall failed as the DUP blocked attempts to nominate a speaker.
Naomi Long said: "I think all of us who sit in that Assembly found this week incredibly difficult. We also raised with him that issue around Dáithá MacGabhann and the implications for him, and others like him, and the transplant waiting list in order that we can find a way through on that issue as soon as possible.
"So, I suppose optimism, but tinged with the reality that there is still considerable work to do."
She added: "I think to some degree it's about listening, it's about trying to sound out what the exact bottom line of some parties is because some have been more cloaked about that than others.
"We have been very clear that our bottom line is about maintaining that dual-market access for Northern Ireland, that uniqueness that would allow us to grow our economy and become successful, and it's important that we're able to do that."
Ulster Unionist leader Doug Beattie claimed that the details on a potential deal were not outlined during his meeting with Mr Sunak.
"We are in that position where we could have something next week, but it may be another couple of weeks yet," he said on the prospects of an agreement.
"We don't know and I don't leave here knowing much more than I knew when we first went in. But certainly the Prime Minister was enthusiastic, engaged and positive, and that has has given me something to take away from this."
Mr Beattie added: "He's saying that things are moving in the right direction. He's confident that when he puts something on the table which is a deal, which he thinks will work for everybody in Northern Ireland, it will be a deal that unionism can accept."
The UUP leader expressed concern that local parties were not being briefed on the details.
"I've been raising this issue many, many times about you must involve the political parties, you must involve business in whatever you're doing, so we can highlight issues before they arise," he said.
"I mean what they think is a good idea we could quite easily say 'that's not going to work in Northern Ireland' and put the red flag up, so we have said that and we said that again that you need to include us.
"There is a nervousness, I would say, between the EU and the UK, that there will be a running commentary on this and that could end up with things backsliding.
"I can understand that but neither can they just dump something in front of us on the table and say 'there you are, there's a deal, take it or leave it'. You know we have to be involved in this and make some common sense decisions."
Want a quick and expert briefing on the biggest news stories? Listen to our latest podcasts to find out What You Need To Know.