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Brexit Minister Lord Frost has said that a deal can be done with the EU over the Northern Ireland Protocol.
It comes after Taoiseach (Irish Prime Minister) Mícheál Martin said there had been a change of tone in the EU-UK negotiations on the protocol.
Irish Foreign Minister Simon Coveney also said issues with the protocol could be resolved "before Christmas".
However, Lord Frost also warned that the UK triggering Article 16 remained a "legitimate option".
Speaking on BBC News NI's Good Morning Ulster programme on Wednesday, Lord Frost said: "I would like to progress this as fast as we possibly can, I'm glad there's ambition on the EU side from what Simon says.
"I think it can be done. Whether it will be done is a different question.
"It's got to be a deal that deals with the problems, solves the underlying problems and makes a difference to businesses and people's lives."
Lord Frost met the Ulster Unionist Party (UUP), the Social Democratic and Labour Party (SDLP) and Sinn Féin on Wednesday.
Sinn Féin junior minister Declan Kearney said his party told Lord Frost to stop "the rhetoric around threats of triggering Article 16".
"This is a time when we now need to see constructive leadership," he added.
Meanwhile, UUP leader Doug Beattie said the protocol was "not a good solution to the problem that we have".
"The protocol might well be benefitting some businesses, and it is and we know that, but others it does not benefit them. The protocol needs fixed."
SDLP leader Colum Eastwood said the UK government was faced with a choice to "continue on with this battle with Europe", or negotiate in "good faith".
"If we were serious about this we'd be trying to resolve this in the next couple of weeks, but it's yet to be determined whether the British government are serious or not."
Lord Frost met the Democratic Unionist Party (DUP) on Tuesday.
He said the issue of how goods move from Great Britain into Northern Ireland is "at the heart" of it.
"We have said that we don't see any reason why goods that everybody acknowledges are going to stay in Northern Ireland need to go through processes," he said.
"Obviously if they go on into Ireland we have said that we will police them in accordance with EU law, that's perfectly reasonable, but goods staying in Northern Ireland - why do they need to go through a customs check?
"We are in the situation where the EU has made some proposals, at the moment without going into detail - I'm not sure they would solve these problems of checks and processes for goods going into Northern Ireland.
"But we keep talking and we're seeing if we can move this forward in a way that does produce consensus."
He reiterated that activating Article 16 was a legitimate option for the UK.
"We're in a negotiation about a very significant real world problem and I hear a lot of concern from business about that real world problem about moving goods and so on," he said.
"It's certainly our preference to try and resolve it without Article 16 but if we can't resolve it in the negotiations Article 16 is a perfectly legitimate option to solve these problems and that's why it has to be on the table."
Asked if the UK would like some sort of light-touch technological solution between Northern Ireland and the Republic of Ireland, he replied: "That's not what we've proposed.
"What we have proposed is an ambitious compromise, it does not require infrastructure or checks at the border between Northern Ireland and Ireland, it involves us accepting responsibility to police goods going into Ireland on the Irish Sea and we think that is a reasonable compromise.
"But the other side of that compromise has got to be that goods move freely into Northern Ireland if they're staying in Northern Ireland."
Speaking in Brussels on Tuesday after a meeting with European Commission Vice-President Maroš Šefčovič, Irish foreign minister Mr Coveney said the EU's hope was that a deal could be done soon but they would see by the weekend if that is "realistic or not".
Mr Šefčovič is due to meet Lord Frost on Friday.
What is the Northern Ireland Protocol?
The protocol is the part of the Brexit deal that prevents a hard Irish border by keeping Northern Ireland inside the EU's single market for goods.
That also creates a new trade border between Northern Ireland and the rest of the UK, something the EU accepts is causing difficulties for many businesses.
It has proposed a package of reforms but the UK is seeking more fundamental changes.
The UK is threatening to use Article 16 of the protocol, which allows for unilateral "safeguard" measures if the deal is leading to serious practical problems or causing diversion of trade.
Those safeguards would amount to suspending parts of the deal, something the EU believes would be unjustified.
On Monday Prime Minister Boris Johnson said: "We would rather find a negotiated solution to the problems created by the Northern Ireland Protocol, and that still seems possible.
"But if we do invoke Article 16 - which by the way is a perfectly legitimate part of that protocol - we will do so reasonably and appropriately."