UK Government sources have cast doubt on the legitimacy of another independence referendum before Brexit - as the battle for Scottish public opinion hots up.

Before the last referendum both the Scottish and UK Governments agreed that it had to be “fair, legal and decisive”.

But Conservative Government sources have now questioned whether a vote before the outcome of Brexit is known would pass the 'fairness' test.

"Would it be ‘fair’ to ask people to make that decision before we know the exact details of the Brexit agreement? That is a big question," one said.

Mrs May has asked Scots to use the upcoming local elections to send a message to the SNP that they do not want a second referendum.

Opinion polls suggest that most Scots do not want another vote any time soon.

But on Tuesday First Minister Nicola Sturgeon accused the Tory Government of using Brexit to attack "the very foundations" of devolution as she dramatically ramped up her rhetoric on independence.

Speculation is growing that Ms Sturgeon could make some kind of announcement on another referendum at her party conference next month.

If she did the Prime Minister faces a dilemma, amid fears that rejecting another vote would increase support for independence.Downing Street has refused to be drawn on whether or not Mrs May would veto a second vote.

The right to hold a legal referendum resides at Westminster, but the Coalition Government temporarily transferred that power to Holyrood in the run up to the 2014 vote.

No 10 insists that the issue is not whether there could be another referendum but whether there should be one and says that the answer to that question is no.

The UK Government is also concerned about fighting on two fronts – battling to keep Scotland in the Union at the same time as haggling with Brussels over Brexit.

A UK Government source added: "Any referendum clearly has to be legal, it has to be decisive and it also has to be fair.”

Ms Sturgeon has said that another vote is "all but inevitable" if Mrs May rejects her plan to keep Scotland in the EU single market.

In a speech to the David Hume Institute in Edinburgh earlier this week she went further, accusing the UK Government of using Brexit to attack "the very foundations" of devolution.

Ms Sturgeon claimed that forces at Westminster were trying to to "rein in" Holyrood and that devolution faced "a grave threat" from Conservative ministers and MPs.

This "harm to Scotland's interests" and identity made an independence referendum "almost a necessary way of giving the people of Scotland a say in our own future direction," she said.

Asked about the possible timescale for a second referendum, she said that her party did not want a situation where Scotland had diverged “hugely” from the EU.

A UK Government spokesman said it had been “very clear” that no powers would be taken away.

At Prime Minister’s Questions, Mrs May said she could not guarantee that decisions that are currently taken in Brussels on agriculture and fisheries would revert to the Scottish Parliament after Brexit.

She said that she was "discussing" that issue with all the devolved administrations, adding that it was important to protect the single market within the UK .

Her answer was in response to a question by SNP Westminster leader Angus Robertson, who said that Scots has been promised control over both areas in the run up to the EU referendum vote.

The UK Government insisted it had not made those pledges, saying they had been promised by some Leave campaigners.

Earlier Scottish Secretary David Mundell had said that Scotland would be handed more powers after the UK leaves the EU.

He offered what he said was an "absolute guarantee" that Holyrood would be given greater control over Scottish affairs after Brexit.

He also told MPs that his Government had guaranteed that all existing powers exercised by the Scottish Parliament would remain the same.