Police Scotland said it was making inquiries into the leak of the document, which suggested Nicola Sturgeon wanted David Cameron to return to Downing Street. Mr Carmichael has admitted he gave permission for the memo to be handed to the press when the General Election campaign was at its height.
A Police Scotland spokeswoman said: “We can confirm that a complaint has been received and enquiries are ongoing to establish whether there is any criminality.”
The Parliamentary Commissioner for Standards, Kathryn Hudson, has confirmed that she has received a number of complaints about the conduct of Mr Carmichael. It is expected that Ms Hudson will decide shortly if an investigation is required.
The Cabinet Office has already completed an inquiry into the leaked memo. It found that Mr Carmichael’s special adviser, Euan Roddin, gave details of the memo to the Daily Telegraph with the permission of Mr Carmichael.
A campaign launched yesterday by Kirkwall-based pro-independence and SNP supporter Fiona MacInnes aims to raise £60,000 “for legal representation to oust Mr Carmichael from his Orkney and Shetland seat. By last night it had raised more than £10,000.
Meanwhile, senior Lib Dem Sir Malcolm Bruce suggested the Commons would be cleared out “very fast” if every politician who had lied was made to quit.
Sir Malcolm made the remarks in defence of Mr Carmichael.
He said: “If you’re suggesting every MP who has never quite told the truth, or indeed told a brazen lie, including ministers, including Cabinet ministers, including prime ministers, we’d clear out the House of Commons very fast, I would suggest.”