The claimant is referred to in court papers only as "CTB", the same initials used in a previous high court hearings against Thomas and the Sun. The defendants are listed as "Twitter Inc, and persons unknown".
The "persons unknown" are reported to be described as those "responsible for the publication of information on the Twitter accounts".
Earlier this month a series of messages were posted on the social networking site claiming to divulge the details of injunctions obtained by several celebrities.
The legal action, which was launched in the high court on Wednesday, is believed to be the first against the US social media site.
No one from Schillings, the law firm representing "CTB", or from Twitter was available to comment.
Thomas and News Group Newspapers, the division of News International which publishes the Sun and the News of the World, were in the high court on Monday in a bid to overturn a privacy injunction preventing a "CTB" from being identified.
But Mr Justice Eady, who issued a temporary injunction in the case last month, ruled against the newspaper, arguing in a written statement that there was "no suggestion of any legitimate public interest in publishing such material".
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