The First Minister, who last night accused Theresa May of “kicking the can” over Brexit following a seven-hour Cabinet emergency summit, flew to the UK capital this morning.
Downing Street has confirmed Mrs May will meet with Ms Sturgeon today.
A time and venue are yet to be confirmed.
Mrs Sturgeon has also met with Labour leader Jeremy Corbyn.
In a tweet following that meeting, the First Minister said she would be “surprised and very disappointed if Labour sold out” for the “bad deal” likely to be available from Mrs May.
The Prime Minister yesterday confirmed she would ask the EU for an extension to the Brexit deadline to “break the logjam” in Parliament.
Mrs May is holding talks with Mr Corbyn to agree a plan acceptable to both, which can be put to the House of Commons ahead of the 10 April summit of the European Council.
She did not originally extend that offer further to include the SNP, which hold the third most seats of any party in the House of Commons.
Ms Sturgeon’s Brexit secretary Mike Russell said the Scottish Government would use the Brexit talks with Mrs May to try to push for voters to be given another say on staying in the European Union.
Mr Russell suggested the “best outcome” to the Brexit deadlock would be a People’s Vote.
While he said Holyrood ministers would see this as a “requirement” in any possible compromise, he was also careful not to close down any options.
Mr Russell told MSPs: “If I were able to wave a magic wand and get what I want – apart from no Brexit, which has been a complete distraction and disaster for the last two-and-a-half years of massively damaging proportions – then it would have to be a very long delay, a referendum, the European elections taking place and perhaps some calmness coming into this to look at the damage that would be done by proceeding along the present lines.”
He stressed “at the present moment it is very difficult to say what will take place in the next 24 to 48 hours”.
Mr Russell said: “I think the best outcome of this would be a People’s Vote on an option.
“But I never will absolutely say never.
“I just do not wish to close any possible avenue in the next 72 hours.
“But I think it would be very difficult to justify refusing to take this to the people.”
He added: “Debate, discussion, negotiation, is really important on a genuine basis, that’s what everybody will be trying to do in the next 72 hours.”
Mr Russell also spoke about the impact the Brexit process has had on politicians, who have been left “wrung out”.
“It is tough,” he said. “People are pretty wrung out by this. The last two-and-a-half years has been very tough for people at the heart of this.
“Everyday you get up in the morning and think; what on earth is going to happen next?”
In a response posted on Twitter last night, Ms Sturgeon questioned Mrs May’s plan after the speech, saying: “This does seem very much like PM kicking the can and, yet again, delaying making any decision that could break her Cabinet.”
“What is missing is an answer from her to the question that many MPs faced up to last night – what is the compromise she is willing to make?”
Ms Sturgeon claimed Mrs May’s position could represent a “trap”.
She posted: “If MPs allow 12 April to pass with no commitment to fight Euro elections, May 22 becomes the inescapable exit day...and PM would then be able to say it’s my deal or no deal. Parliament needs to be very wary about a potential trap.”
The First Minister then added in a third post: “The sensible way forward – and I think the one PM would take if this was a serious attempt to build consensus – is agree to fight election, seek longer delay and allow option of public vote on what Commons agrees.”
MP Nick Boles, who dramatically quit the Tory Party on Monday night, has meanwhile said Mrs May’s top ministerial team was probably the “worst Cabinet collectively” in recorded history.
He told the BBC: “There are some fine people in the Cabinet, genuinely, people who would have been in a Cabinet in any age, but this is the worst Cabinet collectively not only in my lifetime, but I think probably in recorded history.
“Were I still a member of the Conservative Party, one of the contributions I would have made to the debate about the leadership election, that will have to come because Theresa May has announced that she is going to stand down, is that it should not be anyone who is or has been in the Cabinet since 2017.
“None of them in my view has earned the right to lead the country after Brexit. They are all compromised by their collective failure to lead, to unite, to get behind one plan, to sell that plan, to communicate.
“They have all put themselves first, they have all been cowardly when they should have been brave, they have been selfish when they should have been cooperative. None of them should be prime minister after Brexit.”