The force said in a statement last night that it is to hold discussions with the Ibrox club in the wake of Thursday's AGM, which saw finance director Brian Stockbridge and his boardroom colleagues re-elected.
Mr Stockbridge was the focus of most of the anger from supporters at the meeting where four other businessmen, including former director Paul Murray, failed to win election.
Mr Stockbridge received the lowest vote, of 63%, of those re-elected. He is understood to have been on the receiving end of the abusive messages on Twitter.
A Police Scotland spokeswoman said: "We have not received a formal complaint from Rangers Football Club. However, we have contacted the club and will undertake further inquiries to assess if there is criminality involved."