The prime minister dramatically upped the stakes in the fight for the UK's future when she said "now is not the time" for a repeat of the 2014 vote, which the No camp won.
In Edinburgh, Nicola Sturgeon said she would be "up for" a discussion with May about the timing of a new independence referendum.
May went onto attack the SNP's pro-EU position as "muddle on muddle", pointing out that Sturgeon called for the second plebiscite between autumn 2018 and spring 2019 because the United Kingdom voted 52% to 48% for a Brexit, whilst 62% of Scottish voters backed staying in the EU.
"I think we will find the Prime Minister is not in a position she can sustain and I would say to her, in all sincerity here in trying to be consensual, we seem to have a starting point of agreement that both of us think that now is not the right time for a referendum".
As the Prime Minister of this United Kingdom, I will always ensure the voices and interests of England, Scotland, Wales and Northern Ireland are represented as we negotiate to leave the EU.
She continued: "It's now clear, that using Brexit as the pre-text to engineer a second independence referendum has been the SNP's sole objective ever since last June".
Nicola Sturgeon has been urged to withdraw the threat of an "illegimate" second independence referendum amid speculation that the SNP is considering an unauthorised poll.
"Scottish Labour will vote against the SNP's plan for another divisive referendum when a vote is held in the Scottish Parliament next week".More news: Northwestern Beats Vanderbilt for 1st NCAA Tournament Win
Downing Street has rejected the suggestion that the SNP could hold a legal vote without Mrs May's approval.
A third time, Mr Swinney said: 'I think the Prime Minister would be ill-advised to stand in the way of the democratic wishes of the people of Scotland, as expressed democratically in the elected Parliament'.
Also Thursday, Britain's European Union exit moved closer when a bill authorizing the British government to begin divorce proceedings with the bloc became law.
The bill was passed Monday by Parliament but needed royal assent to become a law.
"We are now the party of the new centre-ground of British politics".
The most dramatic option open would be to trigger a snap Scottish parliament election by resigning as first minister, forcing Holyrood to dissolve at a crucial time during the UK's Brexit talks with the EU.
"If Theresa May takes that stance she will be standing in the face of Scottish democracy and that is a very, very unstable and unsafe position for the Prime Minister to be in", Swinney said.
Under the terms of the European Union treaty, Britain will no longer be a member of the bloc two years after May issues the notification.