Nicola Sturgeon said allegations of breaches of election rules by Vote Leave were “deeply and profoundly concerning”.
Speaking at First Minister’s Questions at Holyrood, she also condemned the outing of one of the whistleblowers at the centre of the claims by one of the Prime Minister’s aides, describing Theresa May’s response as “woefully inadequate”.
Vote Leave has repeatedly denied allegations of collusion or deliberate overspending made by former campaign volunteer Shahmir Sanni and former Cambridge Analytica employee Christopher Wylie.
Mr Sanni has also alleged through his lawyers he had been “outed” by No 10 over a relationship he had at the time of the referendum with Stephen Parkinson - a senior figure in the Vote Leave campaign who is now Theresa May’s political secretary - after whistleblowing.
Ms Sturgeon said: “I think many of the revelations we’ve heard, and many of the allegations we’ve heard over the last couple of weeks are deeply and profoundly concerning.
“I thought the outing of a whistleblower by a member of Number 10 staff was utterly disgraceful, and should be condemned by everybody. I think the Prime Minister’s response to that was woefully inadequate.”
“I think there has also been some very serious questions raised over the past number of days about the conduct of the Leave campaign... I think those questions merit very serious and in depth investigation.”
The issue was raised at Holyrood by Scottish Greens co-convener Patrick Harvie, with one year to go until Brexit.
Ms Sturgeon added: “We are today one year from the date when the UK is supposed to leave the European Union, and what I think is utterly inexplicable and shameful is the fact that people today don’t have any more detailed answers about the future relationship with the European Union than they did on referendum day.”