The article further states similar claims against photographer Bruce Weber, who is being accused by more than 15 male models of sexual improprieties.
Testino, whose past subjects include Kate Moss, Madonna, Michelle Obama and members of the Royal Family, was accused by 13 male assistants and models of subjecting them to sexual advances.
Model Ryan Locke worked with Testino on Gucci ad campaigns and called him a "sexual predator".
His photograph of tennis player Serena Williams and her baby daughter appears on Vogue's February edition, unveiled this week - while his other works include the Duke and Duchess of Cambridge's engagement photo and a series of images of Princess Diana in 1997 published in Vanity Fair.
Representatives for Weber and Testino said they were shocked by the allegations.
U.S. photographer Weber, 71, was accused by 15 current and former models of subjecting them to unnecessary nudity and coercive sexual behaviour, according to The New York Times.
In a statement to the paper from his lawyer, Weber said he was "shocked and saddened by the outrageous claims being made against me, which I absolutely deny".
The men recall private sessions in which Weber would ask them to strip down and lead them through breathing and "energy" exercises, in which they would touch both themselves and the photographer. We never had sex or anything, but a lot of things happened.
"Then he crawls on the bed, climbs on top of me and says, 'I'm the girl, you're the boy, '" Locke said.
Weber's racy advertisements for Calvin Klein, Abercrombie & Fitch and other companies helped turn him into a star in commercial and fine art photography.
One anonymous assistant said Testino masturbated on him during a business trip, while another said the photographer groped his backside, the newspaper reported. "I went at him, like, you better get away".
Testino's former assistant, Roman Barrett, accused the photographer of sexual misconduct, describing it as a "constant reality".
Model Bobby Roache said Weber tried to put his hands down Roache's pants during a casting in 2007.
She announced a new worldwide "Code of Conduct" for contributors to all publications of Vogue's parent company, Conde Nast, for which she serves as artistic director.
Lavely & Singer, the law firm representing the pair, has not responded to a request for comment on the claims about Weber from the Press Association and said there would be no additional comment regarding Testino. Recently, photographer Terry Richardson continued to work after being accused in a documentary of sexual assault of female models and denying their claims - until the Weinstein scandal broke.