A small peninsula on Spain's southern coast and a British territory since 1713, Gibraltar is a major point of contention in Anglo-Spanish relations.
It was originally billed as a chance for EU27 leaders to "rubber stamp" two documents - the withdrawal deal and a non-binding political agreement on the UK's future relationship with Brussels.
A Downing Street spokesman said: "She said there had been good engagement between the United Kingdom, the government of Spain and the government of Gibraltar in order to come to an agreement in the withdrawal agreement and associated package of memoranda of understanding relating to Gibraltar".
Spain's demands on Gibraltar are the latest push by European Union states to obtain more on national interests ahead of Sunday's summit of European Union leaders, but diplomats said there was little concern these would scupper prospects for a deal.More news: Russian candidate is front-runner in election for Interpol’s new president
But Borrell rejected this idea, suggesting Sunday's meeting should not take place if the issue were not resolved beforehand.
The EU Commission today agreed the draft text for the future relationship between the United Kingdom and EU after Brexit.
European Council president Donald Tusk said the political declaration, which accompanies the 585-page withdrawal agreement or "divorce" deal, has been "agreed at negotiators' level and agreed in principle at political level". 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.
"As of today, if there are no changes with respect to Gibraltar, Spain will vote no to the agreement on Brexit".
British Prime Minister Theresa May is battling to keep her Brexit deal alive as she readies to return to Brussels for last minute negotiations on Saturday.More news: Facebook Messenger reportedly down around the world
Mr Kurz told the Austria Press Agency in remarks published Thursday that his trip to London is created to help May build support for the deal, which faces opposition in the British Parliament.
"The PM has been clear that we will not exclude Gibraltar, and the other overseas territories and the crown dependences from our negotiations on the future relationship".
Foreign Minister Josep Borrell said yesterday that Madrid needs more clarity on the future status of the Rock before it can offer its full support for the UK's deal.
"I will do everything possible to get an exit agreement", she said.
May won a reprieve from some of her Conservative Party foes after pro-Brexit rebels acknowledged that a bid to trigger a no-confidence vote in May had failed, for now.More news: Netanyahu takes on defence post amid call for early polls
The Irish government has previously said that its priority is to "protect the peace process, the Good Friday Agreement, citizens' rights and the common travel area".