It is alleged that Caroline Farrow, who is also a mother and social commentator, made a transphobic comments during a heated exchange on Twitter last year.
Posting on the social media site, she said: "I now need to inform my children that Mummy is in trouble with the police because there was this girl, who was born in a boy's body, whom I might have inadvertently referred to as a boy."
Surrey Police has been investigating whether a hate crime was committed under the Malicious Communications Act. Such an offence carries a sentence of up to two years in prison.
A row on social media erupted following the 44 year old's appearance on ITV's Good Morning Britain show last September.
Mrs Farrow appeared with the transgender rights campaigner Susie Green to discuss a Girl Guides policy of not informing parents if a transgender youngster joins.
It is in the row which followed that Mrs Farrow allegedly misgendered Ms Green's transgender daughter, Jackie.
Commenting on Twitter, she said: "I probably said 'he' or 'son' or something.
"I have pointed out to the police that I am a Catholic journalist and commentator and it is my religious belief that a person cannot change sex.
If anyone is interested Susie Green is going to be discussing her complaint and my 'distressing spiteful transphobia' on @vicderbyshire this morning. I was invited on the show but declined.— Caroline Farrow (@CF_Farrow) March 20, 2019
"That we are in the middle of a national conversation about what it means to be male and what it means to be female."
Mrs Farrow, who is married to a Roman Catholic priest, says she was told she will be arrested if she fails to attend to the police questioning voluntarily.
Describing the Twitter exchange as "distressing and spiteful", Ms Green filed a formal complaint with Surrey Police, which she later dropped.
I am overwhelmed by media requests but I need to concentrate on a parent consultation about how my son is getting on in pre-school (he has a speech delay) and tonight my daughter is singing the solo in a choral society production of Vivaldi's Gloria & is a bag of nerves. ❤️— Caroline Farrow (@CF_Farrow) March 19, 2019
Appearing on the BBC's Victoria Derbyshire show on Wednesday, Ms Green, who founded a transgender support charity, said she complained about Mrs Farrow because of the overall context of the tweet - not solely because her daughter was misgendered.
She said: "It's not just the misgendering, it's actually the context that she puts it into, and that she calls me a child abuser."
"She constantly refers to my daughter as a boy but that's not...the key issues in here weren't those.
"It was the really damaging things that she said about me and my actions that made me decide that this was an appropriate course of action."
A police spokesman said: "A thorough investigation is being carried out to establish whether any criminal offences have taken place.
"We received an allegation on October 25 2018 in relation to a number of tweets which were posted in October 2018.
"A 44-year-old woman has been asked to attend a voluntary interview in relation to the allegation as part of our ongoing investigation.
"We have been made aware that a number of allegations have since been made on Twitter and we will be making contact with the person concerned in relation to these posts."
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