Twitter has come under increasing pressure to review its abuse-reporting systems after the Labour MP Stella Creasy was threatened with rape for supporting the feminist campaigner Caroline Criado-Perez.

Creasy, the MP for Walthamstow in east London, retweeted some of the messages she had received on the micro-blogging site, adding that such behaviour would have consequences.

"You better watch your back....Im gonna rape your ass at 8pm and put the video all over the internet," read one.

Another said: "If I meet you in an alley you will definitely get fucked."

Creasy used Twitter to inform the police of the threats, warn her abusers that she was logging their threats and taking screen grabs as evidence.

"You send me a rape threat you morons I will report you to the police & ensure action taken," she wrote.

Creasy spoke out as the online petition calling for Twitter to review its procedures attracted almost 60,000 signatures, and a day after police arrested a man in connection with Twitter threats made to Criado-Perez after she succeeded in her campaign to have the image of a woman, Jane Austen, used on a new bank note. The 21-year-old was arrested in Manchester on suspicion of harassment offences following a complaint made on Thursday.

On Monday, Twitter said it had introduced a button for reporting abuse on its latest iPhone app and was looking at expanding this function.

A spokesman said: "The ability to report individual tweets for abuse is currently available on Twitter for iPhone, and we plan to bring this functionality to other platforms, including Android and the web.

"We don't comment on individual accounts. However, we have rules which people agree to abide by when they sign up to Twitter.

"We will suspend accounts that, once reported to us, are found to be in breach of our rules. We encourage users to report an account for violation of the Twitter rules by using one of our report forms."

The response is unlikely to satisfy some critics who have already said the iPhone button is inadequate as it links to a complex form and does not allow reports of multiple abuse tweets.

Criado-Perez, who has received support from MPs and celebrities, said: "It's sadly not unusual to get this kind of abuse but I've never seen it get as intense or aggressive as this.

"It's infuriating that the price you pay for standing up for women is 24 hours of rape threats. We are showing that by standing together we can make a real difference.

"We made the Bank of England change its mind, we can do the same with Twitter."

Criado-Perez, a freelance journalist, organised a campaign which included a petition signed by more than 35,500 people after the Bank of England decided to replace Elizabeth Fry with Winston Churchill on new £5 notes. The move would have meant there were no women apart from the Queen on sterling banknotes.

Her campaign was a success, with the bank announcing last week that Austen would feature on the new £10 when it is introduced in 2017.

Tony Wang, the general manager of Twitter UK, said the company took online abuse seriously.

He tweeted: "We encourage users to report an account for violation of the Twitter rules by using one of our report forms.

"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. We will suspend accounts that, once reported to us, are found to be in breach of our rules."

The shadow home secretary, Yvette Cooper, has written to Wang criticising Twitter's response to the "disgraceful, appalling and unacceptable" comments made about Criado-Perez on the site.

She wrote: "Despite the scale and seriousness of these threats, the official response from Twitter continues to be extremely weak – simply directing Caroline away from Twitter towards the police, and, belatedly, directing users to abuse reporting forms on Twitter.

"Of course it is right to report such abuse to the police, and it is very important that they investigate and pursue this case.

"But social media platforms also have a responsibility for the platform they give users. And in particular they have a responsibility not to tolerate this kind of abuse, rape threats and potentially criminal behaviour."

She added: "The response by Twitter has clearly been inadequate and fails not only Caroline, but many more women and girls who have faced similar abuse on your social network.

"More than 20,000 people have already signed an online petition asking Twitter to allow users to report abuse directly with one click.

"I urge you to go further and ensure that Twitter carries out a full review of all its policies on abusive behaviour, threats and crimes, including more help for Twitter users who experience abuse, a clear complaints process and clear action from Twitter to tackle this kind of persecution."