UPDATE: Twitter has responded to the campaign in an email to Caroline Criado-Perez and Labour MP Stella Creasy.
In an email seen by HuffPost UK, Twitter's Steve Summers asked to have all the police reference numbers and specific tweets sent to him so the company could take action. He said that he would make sure both of the women were "connected to the right people for conversations to continue on Monday".
Creasy had asked for a meeting with Twitter, along with fellow MP Steve Rotherham "on how we ensure that Twitter is able to comply with the Protection From Harassment Act in Britain."
The feminist campaigner who ran the successful bid to get a woman on British banknotes has revealed she has got "up to 50 rape threats an hour" on Twitter.
And prominent journalists, showbiz stars and politicians are rallying to support Caroline Criado-Perez, who runs the Women's Room, threatening to quit the site if nothing is done to stop the abuse.
Criado-Perez said she had been getting the threats for almost 48 hours since the announcement by the Bank of England that it would put Jane Austen on the £10 from 2017.
Caroline Criado-Perez has been subjected to vile abuse on Twitter
She told HuffPost UK: "For me, the really shocking thing is how this has happened over such a tiny, tiny thing. We asked for there to be a woman on a bank note, how does asking that even annoy someone? Annoy someone so much they send a barrage of rape threats? It's kinda gobsmacking."
But Criado-Perez said she was heartened by the response, which she hoped could make a real difference to how hate tweets are reported to the social network.
"I would not say I'm glad it's happened, but if it does have to happen then I'm glad it happened over such a minor, tiny thing.
"That shows it's not about what women are doing, not about feminism. It's that some men don't like women, and don't like women in the public domain."
Tweets to her account, many of which are too grotesque for publication, include one user who said: "Everyone jump on the rape train, @CCriadoPerez is the conductor."
Another wrote: "Hey sweetheart, give me a call when you're ready to be put in your place."
A petition on Change.org has attracted almost 12,000 signatures, calling for Twitter to address the issue.
Kim Graham said she was moved to start an organised petition after seeing the extent of the threats.
"It is time Twitter took a zero tolerance policy on abuse, and learns to tell the difference between abuse and defence. Women standing up to abuse should not fear having their accounts cancelled because Twitter fail to see the issue at hand," she wrote.
Many have rallied online to defend Criado-Perez, including journalists Owen Jones, Hadley Freeman, Laurie Penny and Caitlin Moran, comedian Dara O Briain and Labour MP Stella Creasy and former deputy prime minister John Prescott.
Moran suggested many prominent Tweeters and supporters leave Twitter on August 4th, International Friendship Day, for 24 hours, in solidarity with Criado-Perez and victims of online abuse.
Criado-Perez said she had reported the abuse to the police.
Tony Wang, the general manager of Twitter UK said in a statement on the site: "We take abuse seriously and will investigate reports made via https://support.twitter.com/forms.
"We don't comment on individual accounts, but we have rules which people agree to abide by when they sign up to Twitter. We take online abuse seriously and provide advice and guidance to our users.
"We encourage users to report an account for violation of the Twitter rules by using one of our report form
"Also, we're testing ways to simplify reporting, eg within a Tweet by using the "Report Tweet" button in our iPhone app and on mobile web."
She told HuffPost UK said she would encourage women experiencing online abuse not to follow the policy of "Don't feed the troll" but to call out abusers, and shame them.
"There are so many lovely people who drown them out, so many more of us. I've been told a lot of the men who started being big bullies with the threats have now locked their accounts, that's a win, they had to shut themselves up."