Rumours have been circulating the person is an employee of an insurance company.
In response to queries made by the Herald, a spokesperson for the company said the matter was under investigation.
They said the company expected employees to uphold high standards of behaviour, both inside and outside the workplace.
"Given the matter is under investigation, we cannot make any further comment, however [the company] will take appropriate action if it is found that one of our employees is acting in a way that breaches its Code of Conduct, policies, or otherwise acts inconsistently with our values."
The Herald has chosen not to name the company while it investigates.
Matthews has since filed a complaint with police and Twitter, and it appears the social media giant has removed the comment for violating its rules.
"We want to be accountable and available to the public, but nobody deserves to be threatened," Matthews said.
The councillor has also contacted the insurance company and has informed a senior manager at Wellington City Council.
Matthews said she thought the level of vitriol could be more extreme towards female politicians.
Earlier this year Porirua mayor Anita Baker pulled back from social media following what she described as online sexist abuse.
She decided to no longer engage with constituents on Facebook, saying her male colleagues did not receive the same messaging.
"After a year I'm sick of the abuse that I'm getting and I think being a female mayor and having a female CEO has a lot to do with it.
"I actually am a human being and I have feelings, so I've decided I'm not going to take it anymore."
She said she would continue to use her mayoral Facebook page where she controlled the content.
In another incident this year, a Kāpiti Coast District Councillor was left reeling after offensive sexual remarks about her were posted on a board member's blog.
One piece referred to Jocelyn Prvanov and four other female councillors as "political concubines" of mayor K Gurunathan.
Waikanae Community Board member Geoffrey Churchman, the editor of the site, has since stepped down.
