Scotland First Minister Nicola Sturgeon has said she would consider a second referendum on Scottish independence in the first half of next year if necessary.
Ms Sturgeon told the BBC that could happen if the United Kingdom started the formal process of leaving the EU without Scotland's position being safeguarded.
She has also suggested that Scotland could stay in the UK and the European Union.
However, the UK minister responsible for Brexit, David Davis, said he did not think this would work, although UK Prime Minister Theresa May said she would listen to any options brought forward by the Scottish government.
Following a meeting with Ms Sturgeon on Friday, Mrs May appeared unwilling to consider a second referendum on Scottish independence, saying people in Scotland sent a "very clear message" in 2014.
Mrs May also said she would not trigger article 50 - the formal process of the UK leaving the EU - until there was a "UK approach and objectives".
'Up to the people'
The first minister said this meant Scotland had been put in a very, very strong, strong position.
On the BBC's Sunday Politics Scotland programme, when asked about her position if article 50 was triggered in December and the Scottish government was not "on board", Ms Sturgeon said that was why she was making preparations for a second independence referendum.
"Of course at that point that would be an option and a decision that I would have to consider," she said.
"If you are asking me right now, do I think Theresa May will never ever trigger Article 50 unless I am saying to her I am absolutely happy with the direction that the UK is taking, I don't know that that is the case, but what she did seem to indicate is that she wants, as I want, to see if we can find options that respect how Scotland voted."
When asked if she would be happy to have an independence referendum in the first half of next year, she said: "I will have an independence referendum if I come to conclusion that is in the best interests of Scotland.
"I've always said that. It would be up to Scottish people ultimately to decide if that is right way to go."
She said that if a referendum was going to be held it would make sense for that to happen before the UK left the EU.
Speaking earlier on the BBC's Andrew Marr show about Scotland's position regarding the EU, Ms Sturgeon suggested that Scotland could stay in the UK and the EU.
Ms Sturgeon said they were in "uncharted territory".
"And when you're in uncharted territory with effectively a blank sheet of paper in front of you then you have an opportunity to think things that might previously have been unthinkable and shape the future.
"So I think there are opportunities.
"I think the positive outcome of the meeting I had with the prime minister on Friday was that she said she was prepared to listen to options that the Scottish government would bring forward to give effect to how Scotland voted and we will certainly bring forward options.
"Let's see what progress we can make."
The first minister said that in Brussels she had encountered "a warmth, an openness a great sympathy to the position Scotland finds itself in".
She said Mrs May's comments that article 50 would not be triggered until all parts of the UK agreed, left Scotland in a "very, very strong position".
"That's a position I'm going to use as well as I can," she said.
Earlier this week, Scottish Secretary David Mundell said the idea of Scotland staying in the UK and the EU was "fanciful".
And speaking to Sky, David Davis, the UK government minister responsible for Brexit, said he did not think Scotland could have such an arrangement.
He said no-one could have a veto over Brexit, saying the government would not ignore the referendum result.
"One of our really challenging issues to deal with will be the internal border we have with southern Ireland, and we're not going to go about creating other internal borders inside the United Kingdom.
"They [Scotland] can't have a veto because 17 and a half million people have given us a mandate, they've told us what to do, we can't disobey it."
Mr Davis said he wanted to trigger Article 50 "early next year" and the timetable had to allow for the UK's "huge negotiation" and for International Trade Secretary Liam Fox to strike deals around the world.
"We'll get a very, very, very large trade area, much, much bigger than the European Union. Probably 10 times the size, well I'm not going to set Liam targets, but a multiple of the size of the European Union. And the moment we leave, they'll all come in."
Mr Davis said he did not believe Scotland would want another independence referendum.
Following a "positive" meeting with Ms Sturgeon in Edinburgh on Friday, Mrs May said: "I'm willing to listen to options and I've been very clear with the first minister today that I want the Scottish government to be fully engaged in our discussion.
"I have already said that I won't be triggering Article 50 until I think that we have a UK approach and objectives for negotiations - I think it is important that we establish that before we trigger Article 50."