Austria now holds the rotating European Union presidency, and Kurz told the Austria Press Agency that his visit is meant to help the prime minister bolster support for her deal and get a realistic idea of how strong support for it is.
British Prime Minister Theresa May says the agreement with the European Commission on the draft political framework on a future relationship represents the "right deal" for Britain.
It includes references to Britain developing "an independent trade policy" and declares an end to free movement of citizens between Britain and the European Union - two key demands of the Brexit campaign in the 2016 referendum.
Spain's prime minister, Pedro Sanchez, has said his government is "annoyed" over the issue of Gibraltar, a British Overseas Territory on the Spanish coast, and France is understood to have sought amendments to the wording on fishing rights in United Kingdom waters.
"In parliament there's a lot of focus on who's going to vote for the deal or not, and outside I think people are thinking: "Actually, let's make sure we can get this through and get on with delivering".
The Prime Minister updated around 50 representatives from Northern Ireland businesses, universities and the voluntary and community sector on the draft Brexit deal at a reception, also attended by David Lidington, the Chancellor of the Duchy of Lancaster, and the Business Secretary Greg Clark.
The political declaration is not legally binding and sets out both sides' aspirations for what a future relationship will look like rather than a conclusive trade deal.
She said businesses would benefit because Northern Ireland would "be a gateway to both the European Union market and the rest of the U.K.'s market".
Northern Ireland's Democratic Unionist Party (DUP), on whom May's Conservatives depend for their majority, has been flexing its muscles, fearing that efforts to avoid a hard border with Ireland will lead to the province drifting apart from the rest of the UK.
Spain's stance is opposed by both London and Belfast, where Irish Foreign Minister Simon Coveney said a deal that took almost two years to negotiate "should not be changed now".
Downing Street dismissed any suggestion of reopening talks over the UK's draft withdrawal agreement ahead of Sunday's summit, amid Spain's demands.
Pro-Brexit Conservatives have also savaged the divorce deal, which they say keeps Britain too close to the EU.
Spain has long claimed sovereignty over the territory, which adjoins the Spanish mainland but has been a British territory since 1713.
But late Thursday Sanchez maintained his threat to scupper the deal. "My government will always defend the interests of Spain", Sanchez wrote on Twitter.
"We had to do something to show our displeasure", DUP Brexit spokesman Sammy Wilson told the BBC.