Feargal Dalton, an SNP councillor in Glasgow, was interviewed several times by officers about a series of tweets which referred to him as "Paddy" and accused him of glorifying terrorism.
The messages also mentioned well known bars in Glasgow's west end which the Irish-born councillor has frequented in the past.
But several weeks after Glasgow City Council contacted the police and Mr Dalton was spoken to, Councillor Dalton said he was told Twitter would not divulge the details of the individual making the comments.
He said: "I received racial abuse on social media and brought it to the attention of my bosses in the city council. They contacted the police who took a statement from me. The police subsequently contacted me to say they could not proceed with the case because Twitter will not disclose who owns the social media account in question."
In one of the tweets sent in June the poster accuses Mr Dalton, who represents Partick West, of drinking in pubs frequented by Celtic supporters "singing your IRA tunes and celebrating the deaths of British soldiers".
In another to Mr Dalton, an officer in the Royal Navy for more than a decade, the same tweeter posts a picture of a British Army officer with the words: "And you live under it Paddy and we will never give it up."
Although not part of the complaint to Police Scotland Mr Dalton received further tweets at the weekend referring to him as "Paddy" and making reference to the IRA.
SNP sources said while Mr Dalton was "a resilient person" who would "not have sleepless nights" about the online abuse it clearly amounted to racism.
One said: "What's going on here is racially motivated. Call it what it is. Does this mean that if we want to racially abuse people, as long as we can do it online you should get away with it? You wouldn't get away with doing it on the street, so why does it seem you can act with impunity on Twitter."
A spokesman for Twitter said: "We're looking into this. We will let you know when we have more information, though we may not be able to comment on an ongoing investigation."
The spokesman referred to the company's policy on divulging personal details of users which states: "Non-public information about Twitter users will not be released to law enforcement except in response to appropriate legal process such as a subpoena, court order, or other valid legal process or in response to a valid emergency request."
A Police Scotland spokeswoman said inquiries were ongoing but refused to comment on Mr Dalton's claims it had been refused information by Twitter.