Lothian and Borders Police said there are "inquiries" ongoing into SNP MSP Christine Grahame.
The Herald reported yesterday the Presiding Officer has written to Ms Grahame on behalf of the corporate body "conveying its strong disapproval" about misuse of resources.
The Scottish Parliament Corporate Body (SPCB) did not uphold all the allegations made by her former employee, Mark Hirst, but found three emails indicated a misuse of resources.
Ms Grahame said on Wednesday she was pleased the SPCB was taking no further action.
A spokeswoman for Ms Grahame said: "Mr Hirst is a former employee who has admitted himself that he is 'disgruntled', who secretly recorded his employer and who was dismissed on the ground of gross misconduct. She is pleased that his complaints have been repeatedly rejected.
"Mr Hirst has now indicated his intention to take his complaints to an Employment Tribunal."
The Crown Office said the matter remains under consideration.