Gareth Compton, Birmingham City councillor for Erdington, left a message online after Yasmin Alibhai-Brown spoke on Radio 5 Live's breakfast show on Wednesday, talking about human rights in China.
reported that Compton allegedly tweeted: "Can someone please stone Yasmin Alibhai-Brown to death? I shan't tell Amnesty if you don't. It would be a blessing, really."
After police were informed of the comment, Compton was arrested under the Communications Act 2003 and bailed.
He has also had his membership in the Conservative Party indefinitely suspended, pending further investigation.
"I made an ill-conceived attempt at humour in response to Yasmin Alibhai-Brown on Radio 5. I [apologise] for any offence caused, it was wholly unintentional,” the BBC
quoted him as saying.
Alibhai-Brown said that her daughter had seen the tweet earlier but had not said anything, although it had upset her.
She had not known what was online until she received a call.
"If I, as a Muslim woman, had said about him what he said about me then I would be arrested in these times of the war against terror," she said.
"He does not have more of a right to say these things about me that I do about him and I think words matter when you are in public life."
Roger McKenzie, Unison's West Midlands regional secretary, said that he received so many complaints that he called on Compton to resign from the council.
"Birmingham is a multicultural city and the council's workforce reflect this,” he stated
"It is clear that Councillor Compton is out-of-touch with both his city and the council staff.
"It is wholly unacceptable for a public official to make such racist comments. Councillor Compton must resign his seat immediately."