No details were given of the incidents which reportedly involved local SNP councillors.
The investigation was revealed as it emerged new Midlothian councillors are to be given extra advice on personal safety in the wake of the death of MP David Amess.
Councillors unanimously backed a motion calling for elected members to receive the same support and security as national politicians.
Councillor Joe Wallace (SNP), who brought the motion, told a virtual meeting of the council: “We as councillors are just as vulnerable as MPs and MSPs.
“We are easily recognised in the streets because we gladly put our photographs through people’s doors when it comes to election time, we can be reported in the media and words can be, and have been, twisted and taken out of context for political gain, and for modern methods of communication can be targeted at a particular audience or group.
“Unfortunately we have radicalised, hateful, bigoted people amongst us and when the rich and the powerful are seen to act with impunity they think they can do likewise.”
Cllr Wallace said he and his SNP colleagues Councillor Debbi McCall and MSP Christine Grahame had all been targeted by “verbal abuse, harassment and attempts at intimidation”.
He said: “I cannot say too much about it as it is in the hands of the procurator fiscal, however I have particular reason to thank Police Scotland and the fire rescue for providing me with particular advice on personal safety.”
Councillor Janet Lay Douglas (Con) told of her experience not long after being elected in 2017.
She said: “As a new councillor I experienced in the early days a stalking incident which I left too long. It was about six months before I raised the issue with the council.
“I do feel if there are any new councillors coming in it would be beneficial to have something there on how to protect themselves”
Council officers agreed to include additional personal safety advice in new member packs which will be handed out following the council elections next year.
And Councillor Stephen Curran (Lab) backed the motion brought by Cllr Wallace and seconded by Councillor Cath Johnstone (SNP), saying all councillors should think about what they post online.
He said: “Most of us probably spend more time face to face with our constituents than MPs or MSPs and sometime we taken it for granted. This is a good time to reflect on that.
“Councillor Wallace raised a good point about social media headlines and political gain and I don’t exclude myself from this when I say we all need to think about what we post on social media more going forward.”
The motion, supported by councillors, moved that “any local councillor, either serving or retired, who is either under threat or feels under threat, has their property attacked or has their surgeries disrupted by anybody, be given the same level of protection and the force of the law applied to any perpetrator and that any such person be subject to the same penalties as any person who attacks any MP or MSP.”
It added: “We ask that the Chief Executive writes to the Justice Secretary and asks that he takes cognisance of the fact that we are also elected by members of the public and deserve nothing less.”