Miss Church and her parents had until now refused to accept an offer of damages and costs from News Group Newspapers, which owned the now-defunct tabloid, and the case had been due to be heard on Monday.
But sources close to the case said yesterday that Miss Church was “close to settling” her case after NGN made an offer of between £380,000 and £500,000 to pay the family damages and legal costs.
Miss Church and her parents were the only plaintiffs not to have settled their claims out of an initial batch of 60 who sued NGN for breach of privacy.
A court case would have been particularly badly timed for NGN’s owner Rupert Murdoch, who announced on Monday that the first edition of a Sunday edition of the Sun would appear this weekend.
The damages element of the settlement is understood to be more than £180,000, which would average £60,000 for each of the three claimants, making it one of the higher awards agreed this year.
Miss Church, 26, her mother, Maria, and her stepfather, James, had complained that the News of the World published distressing stories about them that had originated from voicemail messages hacked by Glenn Mulcaire, the private detective jailed for phone hacking in 2007.
A spokesman for News International, the parent company of NGN, declined to comment.