UK Prime Minister Theresa May and Scottish First Minister Nicola Sturgeon met for an hour at a Glasgow hotel, days after Sturgeon demanded a referendum on Scottish independence to be held within two years.
May's London-based government would have to approve a legally binding referendum, and May says "now is not the time" for a new independence vote.
Sturgeon said the meeting had been "perfectly businesslike and cordial," but had not broken the logjam.
Tomorrow, Scotland's Edinburgh-based parliament is due to vote on a motion formally authorizing Sturgeon to seek a new referendum.
Asked what would happen if her call for another referendum was rejected by the British government, Sturgeon said: "I will set that out in due course."
May is touring the UK before triggering the EU exit process, trying to shore up support for Britain's exit from the bloc.
In a speech to civil servants at the Department for International Development near Glasgow, May promised stronger powers for Scotland, Wales and Northern Ireland and said she would never allow "our union to become looser and weaker, or our people to drift apart."
"As Britain leaves the European Union, and we forge a new role for ourselves in the world, the strength and stability of our Union will become even more important," she said.
May also said today that "now is the time when we should be pulling together, not hanging apart. Pulling together to make sure we get the best possible deal for the whole of the U.K."
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