The BBC says that details of the settlement will be revealed at London's High Court on Monday, when the two-week trial was due to commence.
Church and her family were the only party out of around 60 hacking cases unable to reach a settlement deal with News Group Newspapers, the parent company of the now defunct News of the World.
Her case was due to go to court just a day after Murdoch's firm re-enters the Sunday tabloid market with a seventh-day edition of The Sun, providing a potentially embarrassing situation for the publisher.
Church claims that the phone hacking, conducted by private detective Glenn Mulcaire on behalf of the News of the World, led to the publication of stories that caused immense harm and distress to her family and personal life.
Speaking at the Leveson inquiry into press ethics last year, the singer said that her mother attempted suicide after finding out that the News of the World had run a story about her husband's affair.
No details are known about the settlement, although the Financial Times claimed this week that Church, along with her mother Marie and stepfather James, were set to accept between £380,000 and £500,000 in damages and costs from NGN. The damages element is thought to be around £180,000.
Comedian Steve Coogan and Tony Blair's former spin doctor Alastair Campbell have already agreed "substantial" settlements with Rupert Murdoch's news group, while Sienna Miller accepted £100,000 from the publisher.
The phone hacking scandal, which hit the headlines last year after revelations that the News of the World targeted the phone of murdered school girl Milly Dowler in 2002, has left parent company News International's reputation in tatters.
There are thought to be at least 50 other hacking cases in the pipeline, but the wide-ranging admissions of culpability by NGN have meant that the only legal issue remaining is how much compensation should be paid in each case.
Meanwhile, Cherie Blair yesterday launched legal proceedings "in relation to the unlawful interception of her voicemails".
The wife of former prime minister Tony Blair is thought to be suing News Group Newspapers, although a spokesman for News International declined to comment.