They were asked to investigate after Ms Davidson said pro-union supporters at sample openings of ballot boxes took tallies of postal votes before the count.
It is understood the Tory leader was spoken to yesterday as a witness and party sources say there is no suggestion of any wrongdoing on her part, according to the Herald newspaper.
A spokesman for the party said: “Ruth had arranged to speak to police in her office to help them with complaints they had received.”
Speaking on BBC Scotland after the polls closed on September 18, Ms Davidson told the Scotland Decides programme: “We have had people at every sample opening around the country over the last few weeks while that’s been coming in and we have been incredibly encouraged by the results from that.
“Going into today, from the postal votes that were cast, our side would have had a lead and I think that we have a confidence, I hope a quiet confidence, that the quiet majority of Scots have spoken today.”
She said postal votes were not counted until after the polls closed but added: “Different local authorities have had openings around the country. It is illegal to discuss that while any ballot is ongoing, so until 10 o’clock tonight no-one could talk about it.
“But there is people in the room that have been sampling those ballot boxes as they have been opened and they have been taking tallies and the reports have been very positive for us.”
Postal ballot openings are held to verify the ballots are genuine and that the signature and date of birth given along with the ballots match official records.
The Scottish Independence Referendum Act 2013 states that ballots must be kept face down during the process and precautions must be taken to prevent anyone from seeing the votes cast.
Agents for the campaigns are allowed to attend but are bound by the Act not to “attempt to ascertain at the proceedings in connection with the receipt of the ballot papers the outcome for which any vote is given in any particular ballot paper or communicate any information with respect thereto obtained at those proceedings.”