Nicola Sturgeon has suggested she could form a "coalition" with like-minded ministers across the UK to help secure a "soft exit" from the EU.
Scotland's First Minister - who strongly backed a Remain vote in the referendum - wants the UK to remain part of the European single market as the "least-worst position" of the leave negotiations.
She has previously described a future outside the single market with only limited access and significant restrictions on free movement as a "hard Brexit" scenario and is now ready to work with other UK ministers concerned about the consequences of leaving the market.
In an interview with BBC Scotland, the First Minister said: "Can there be a coalition across the UK that gets the UK into a more sensible position?
"I think that's worth a good try because Theresa May, she was on the Remain side, so presumably she knows the real risks of removal from the single market.
"Let's have a try at getting the UK, not into the best position - because the best position in my view would be continued membership of the EU - but let's try and get the UK as a whole into the least-worst position and that means staying in the single market."
Michael Russell, the recently-appointed minister for UK negotiations on Scotland's place in Europe, said the UK Government needs to make its position clearer.
He told BBC Radio Scotland's Good Morning Scotland programme: "We don't precisely know as yet what the debate was at Chequers but it is quite clear that there are ministers who have concerns about this.
"There were ministers who voted to Remain and there are ministers who are saying to themselves ... this could be very detrimental.
"This is the issue between hard Brexit and soft Brexit.
"This is not about accessing a shop, this is signing on to the four freedoms, the things that underpin the single market, and unless you sign on to those you do not get to access the single market and that could be - many are now saying - very difficult."
Scottish Secretary David Mundell said Ms Sturgeon will have a "very important role" but the UK Government will lead on EU negotiations.
He told BBC Scotland: "The UK Government is responsible for Scotland's membership of the EU and for foreign affairs, so obviously the UK Government is going to take the lead in the negotiations in relation to our position in the EU.
"We've got to formulate a position which represents the best interests of Scotland, the best interests of the United Kingdom.
"Of course the first minister, responsible for devolved issues in Scotland, has a very, very important role."
Scottish Liberal Democrat leader Willie Rennie said: "Nicola's big idea is forming a coalition with Conservatives in London. We wish her well with that.
"Does the First Minister really expect us to believe that she can reach agreement with Tories in London when she has trashed the agreement we reached in Scotland over Brexit?
"An obsession with independence makes her incapable of reaching a constructive agreement with anybody."
Brexit Secretary David Davis told the Commons he would consider plans which offer separate arrangements between Scotland and the EU.
But he added he does not think such proposals would work.
SNP MP Tommy Sheppard (Edinburgh East) told Mr Davis: "We were promised in 2014 that Scotland would be respected within this UK.
"So can I ask you if proposals emerge in the months ahead which offer the prospect of separate and different arrangements between Scotland and the European Union, will you listen to them and consider them in good faith or will you reject them out of hand?"
Mr Davis, in his reply, said: "The Prime Minister said we'll look at all proposals.
"Now the one you suggest, I can't see how it would work, I really can't, but we'll look at it.
"I have to say to you, right up front, as I said to the First Minister when I spoke to her about it, I can't see how that can be made to work."
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