Police are looking into remarks by UKIP candidate Carl Benjamin after Labour MP Jess Phillips accused him of malicious communications.
Mr Benjamin, who is standing in the European elections, tweeted that he "wouldn't even rape" Ms Phillips.
He has refused to apologise for the remark made in 2016, arguing that "any subject can be the subject of a joke."
Ms Phillips told the BBC's Victoria Derbyshire programme she "cried in the street" after hearing a video by him.
She said that until then she "had been putting a brave face on it and pretending that it was all fine and that I could cope".
The Birmingham MP has called for people who "promote rape and sexual violence" to have a lifetime ban from running for elected office.
After the interview, UKIP's official Twitter feed posted: "The year is 2019, jokes you'd hear down the pub are now worthy of police investigation."
West Midlands Police said it had received a report of malicious communications relating to Mrs Phillips.
A spokesman said: "Officers have spoken to Ms Phillips and the comments are being investigated to establish if an offence has taken place."
The MP for Birmingham Yardley told Victoria Derbyshire she did not fear for her physical safety, but worried for her mental health after thousands of messages from Twitter users attacking her in the last year alone.
"Sometimes I would rather someone punch me in the face than the constant degradation you suffer as a woman in the public eye," she said. "It is constant, it constantly belittles you, it makes you blame yourself."
On Mr Benjamin, she said she could not understand how a person who wrote the comments online was allowed to run to be an MEP.
"It was one thing when he was just some idiot off the internet with a load of bros following him," she said.
"It is a different thing when he is standing on the same platforms that I am standing on, that he will potentially go to a Parliament himself as an elected representative when he has said these things.
"I just cannot believe our system is so weak at the moment that that's allowed to happen."
What disqualifies you from standing as an MEP?
To be able to stand as a candidate in a European election in the UK, you must be at least 18 on the day of your nomination, and be either a British or Irish citizen, an eligible Commonwealth citizen, or a citizen of any other member state of the European Union resident in the UK or Gibraltar.
However, you cannot stand if:
- You have been detained in prison for more than a year
- You have been convicted or reported guilty of a corrupt or illegal electoral practice
- You are currently subject to a bankruptcy restrictions order or debt relief restrictions
Source: Electoral Commission
Ms Phillips has spoken out on behalf of rape and domestic violence victims in Parliament, and worked for the charity Women's Aid.
Mr Benjamin made the initial comments in reply to a tweet from her about rape and death threats sent to women online. His tweet, which said, "I wouldn't even rape you... feminism is cancer," has since been deleted.
The MEP candidate for the South West refused to apologise for the post at news conference in London last month, accusing media outlets of "smearing" him.
In a video posted later that month - which is still available online - he said: "There's been an awful lot of talk about whether I would or would not even rape Jess Phillips... I suppose with enough pressure I might cave, but let's be honest, nobody's got that much beer."
Mr Benjamin told Buzzfeed News it was humour, adding: "Any subject can be the subject of a joke. The alternative is a world devoid of humour, the essential tool we use to reduce the horror of events that are beyond our control."
Ms Phillips said she broke down in the centre of Birmingham after hearing the video.
Members of UKIP's Swindon branch said Mr Benjamin had only joined the local party recently and he should be deselected.
But the party's leader, Gerard Batten, defended his candidate, describing him as a "classical liberal" whose message had been "satirical" in nature.
The programme contacted Mr Benjamin for comment, but he did not return its request.