The tweet, which appeared on the feed of Nick Harrington, who sits on Warwick District Council, contained offensive and racist language towards the Irish.
His account has since been deleted, but a screen grab of the tweet has been widely re-circulated, the Coventry Telegraph said.
The BBC has attempted to get a comment from Mr Harrington.
Insp Russell Barker, of Warwickshire Police, confirmed police were investigating and treating it as a suspected hate crime.
"It is an ongoing investigation at the moment. Until we complete the investigation we cannot comment further," he said.
Mr Harrington represents the Stoneleigh & Cubbington ward.
Council leader Andrew Mobbs said he was "horrified" by the tweet and had suspended Mr Harrington from the Conservatives on Warwick District Council for six months.
"I can confirm that I have this morning suspended Councillor Harrington from the Conservative group for a period of six months and there will also be an investigation," he said.
"This behaviour is completely unacceptable. I have tried to speak to him but I have had to notify him by email.
"This councillor plays no part in the views, policies or thrust of our group," Mr Mobbs added.
He said a formal investigation would now take place through the council's standards procedure.
A councillor has been suspended over a racist tweet sent from his account during the Eurovision Song Contest.