Anna Wintour, the mag's editor and artistic director of publishers Conde Nast, and Robert A Sauerberg Jr, its CEO, told the paper the company was "deeply disturbed" by the claims.
Representatives for both photographers said they were dismayed and surprised by the allegations.
"I'm completely shocked and saddened by the outrageous claims being made against me, which I absolutely deny", Weber said in a statement from his attorney.
"Like my agency said, he has a lot of power". The allegations span from the 1990s to the present day, and allege that both photographers used their standing in the fashion world to sexually manipulate, molest, abuse, and coerce the male models they worked with. Meanwhile, the law firm that represents Testino has claimed that former employees were "shocked by the allegations" and "could not confirm any of the claims".
DIANA's favourite photographer Mario Testino has been accused of molesting several male models - claims he strongly denies.
He recently photographed the February cover of Vogue, featuring Serena Williams and her daughter and received an Order of the British Empire in 2014.
Ryan Locke, a model who worked with Testino on Gucci campaigns, described him as a "sexual predator". Tillman said Testino's brother came into the room and made the photographer get off Mr. Tillman.
One assistant, who wished to remain anonymous, said Testino masturbated on him during a business trip, while another said he was groped on the backside, the paper said.
"I remember him putting his fingers in my mouth, and him grabbing my privates", another model, Robyn Sinclair, told the The Times about Weber.
Another model described Testino as "a mentor who took it a step too far" after alleging he put his hands down his pants and turned up at his hotel room asking for sex.
"We never had sex or anything, but a lot of things happened - a lot of touching, a lot of molestation".
Model Josh Ardolf claimed that Weber grabbed his genitals during a nude shoot.
Men said to the paper that they had asked to participate in "breathing exercises" which could include inappropriate touching, while model Bobby Roache said Weber tried to put his hands down Roache's pants during a casting in 2007.
Wintour published an op-ed in Vogue.com in which she announced that Condé Nast would be establishing a new Code of Conduct. "Today, allegations have been made against Bruce Weber and Mario Testino, stories that have been hard to hear and heartbreaking to confront". In addition to the Weitzman brand, the Condé Nast organization has severed ties with both Testino and Weber, according to reports from FN's sister publication WWD.
All models appearing in Conde Nast-commissioned fashion shoots must now be 18 or older, according to the new rules.
"Recreational drugs are not permitted", and any shoot involving nudity, skimpy clothing or sexual poses "must be approved in advance by the subject".