Associated Newspapers, publisher of the Daily Mail, The Mail on Sunday and the Mail Online, one of the most widely read news websites in the world, said it "utterly and unambiguously" denied the allegations.
The others involved in the litigation are the actresses Elizabeth Hurley and Sadie Frost, Elton John's husband and filmmaker David Furnish, and Doreen Lawrence, the mother of Black teenager Stephen Lawrence who was murdered in a racist attack in 1993.
The individuals had become aware of "highly distressing" evidence revealing they had been victims of breaches of privacy by Associated Newspapers, law firm Hamlins said in a statement.
It said the breaches included placing listening devices inside people's cars and homes, commissioning the bugging of live, private telephone calls, payment of police officials for sensitive information, and impersonating individuals to obtain medical records.
"They have now therefore banded together to uncover the truth, and to hold the journalists responsible fully accountable, many of whom still hold senior positions of authority and power today," Hamlins said in its statement.
Hamlins said it was representing Harry, younger son of King Charles, and Frost, while Lawrence, Hurley, John and Furnish are being represented by the law firm gunnercooke.
"We utterly and unambiguously refute these preposterous smears which appear to be nothing more than a pre-planned and orchestrated attempt to drag the Mail titles into the phone hacking scandal concerning articles up to 30 years old," a spokesman for the publisher said.
"These unsubstantiated and highly defamatory claims - based on no credible evidence - appear to be simply a fishing expedition by claimants and their lawyers, some of whom have already pursued cases elsewhere."
Harry, the Duke of Sussex, has already brought a number of lawsuits against Associated Newspapers' publications.
He is currently suing the Mail on Sunday for libel over an article which stated he had tried to keep secret details of his legal fight to reinstate his police protection, and last year won damages from the same paper over claims he had turned his back on the Royal Marines.
His wife Meghan also won a privacy case against the publisher last December for printing a letter she had written to her estranged father.
The couple's relations with Britain's tabloid press collapsed following their marriage in 2018, and they have previously said they would have "zero engagement" with four major British papers, including the Daily Mail, accusing them of false and invasive coverage.
Media intrusion was a major factor they cited in their decision to step down from royal duties and move to the United States two years ago.
Elton John also defended the couple himself after newspapers accused them of hypocrisy for using his private jet for a flight to stay at his home in the south of France while calling for action to tackle climate change.
A spokesperson for the couple said they had no comment beyond the Hamlins' statement.
Others involved in the action have also previously brought legal action against major media organisations. Frost was awarded 260,250 pounds ($290,595.15) compensation in 2015 after she and seven other celebrity figures sued Mirror Group Newspapers for hacking messages on their phones.
Hurley, John and Furnish also settled phone-hacking claims against News Group Newspapers – publisher of the now-defunct News of the World – shortly before trial in 2019.
An eight-month criminal trial into hacking at the News of the World in 2014 resulted in the conviction of former editor Andy Coulson, who later went to work for then-Prime Minister David Cameron as his communications chief.
($1 = 0.8956 pounds)