SNP Westminster group leader Angus Robertson called on the Prime Minister to offer a compromise to “protect Scotland’s place in Europe” in exit negotiations.
His comments suggest a commitment from Mrs May to try to keep Scotland in the European Union, or at least the single market, would be enough to avoid a second independence vote.
But an SNP source denied reports that Nicola Sturgeon would attempt to get an independent Scotland into the European Free Trade Association (EFTA), which includes Norway and Iceland and participates in the single market, rather than rejoin the EU as a full member.
Mr Robertson told the Guardian: “There may only be days, may only be weeks, but where all of our efforts are currently focused is trying to convince the UK Government to come to a compromise agreement protecting Scotland’s place in Europe.
“If that road runs out and if we have to have that referendum, we will be turning our attention to making sure that we are making the case publicly, intellectually and in every other way so people understand the choice of a hard Tory Brexit Britain or a Scotland able to maintain its relations with the rest of Europe.”
Mr Robertson went on: “If the UK Government genuinely believes in a United Kingdom (it must) take the needs, interests, concerns of the different parts of the UK seriously.
“The Tories are boxing themselves into a very dangerous corner.
“For a party that claims to be a unionist party they are making it very difficult for people in Scotland, who are not traditionally SNP voters, to look to the future of a Tory-run Britain and accepting that as our best way forward.”
Scottish Conservatives leader Ruth Davidson hit back, telling the newspaper: “These are surreal comments.
“How can he say the SNP is focused on negotiations with the UK Government when Nicola Sturgeon has just broken off those talks to unilaterally declare another divisive referendum on independence?
“Everyone knows where the SNP has invested all its attention since the EU referendum - in trying to break up Britain.”
The row comes ahead of an appearance on Wednesday by Brexit Secretary David Davis at a Commons select committee session on the UK’s negotiating objectives for its withdrawal.
Mr Robertson’s comments also come after Spain’s foreign minister Alfonso Dastis suggested Madrid would veto an independent Scotland joining the union.
Any prospective application to the EU can be blocked by any member and Spain is nervous about its own Catalan and Basque separatist movements.
And the Dutch prime minister Mark Rutte warned Brexit could cause the UK “irreparable harm”, saying growth had only remained good as a result of a weak pound.
On Tuesday, Ms Sturgeon warned Mrs May she should not block a new independence referendum and that the Scottish Parliament must determine the “timing, franchise and the question”.
The Scottish First Minister also insisted she had a “cast-iron democratic mandate for an independence referendum”.
She made the comments after Mrs May accused the SNP of “playing politics with the future of our country” and said a second referendum would create “more uncertainty and division”.
Mrs May’s comments were seen as an indication she could delay the referendum until after the expected completion of the Brexit process in spring 2019.
Some reports have suggested permission could be conditional on the SNP gaining an absolute majority in the 2021 Scottish elections.
Ms Sturgeon responded to the Prime Minister, pointing out Mrs May took the keys to Number 10 without an election or even completing a Tory leadership contest.