A series of claims were made against the pair in a New York Times article.
Weber denies the claims, while lawyers for Testino said his accusers "cannot be considered reliable sources".
Vogue publisher Conde Nast said it would not work with the men "for the foreseeable future".
Artistic director Anna Wintour, who is also Vogue's editor-in-chief, said in a statement that the claims had "been hard to hear and heartbreaking to confront".
In a separate statement, Wintour and Conde Naste chief Bob Sauerberg also said they were "deeply disturbed" by the accusations.
Other Conde Nast titles include GQ and Vanity Fair.
In a statement to the New York Times from his lawyer, Weber said he was "shocked and saddened by the outrageous claims being made against me, which I absolutely deny".
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.
The paper reported that the legal firm representing Testino, 63, said those complaining of harassment could not be considered reliable, and that his lawyers had spoken to several former employees who were "shocked by the allegations" and "could not confirm any of the claims".
The allegations against Testino date back to the mid-1990s and include groping and masturbation, the paper reported.
Ryan Locke, a model who worked with Testino on Gucci campaigns, described him as a "sexual predator".
Hugo Tillman, a photographic assistant, said Testino pinned him down on a bed before being removed by another person.
Another assistant, Roman Barrett, said sexual harassment was a "constant reality".
He claims the photographer masturbated in front of him.
In 2014, the Peruvian photographer was made an honorary OBE for his services to photography and charity.
Testino has also long had ties to the Royal Family, photographing Diana, Princess of Wales, and more recently the Duke and Duchess of Cambridge on their engagement and later, for the christening of Princess Charlotte.
He has worked with brands such as Burberry and Michael Kors and his photographs of Serena Williams and her newborn baby feature on the cover of the February edition of US Vogue.
US photographer Weber, 71, is accused by 15 current and former models of subjecting them to unnecessary nudity and coercive sexual behaviour, according to the New York Times.
Weber's work includes campaigns for Calvin Klein, Ralph Lauren and Abercrombie & Fitch as well as several films.
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 trousers during a casting in 2007.
In her statement on the Vogue website, Wintour said: "Both (photographers) are personal friends of mine who have made extraordinary contributions to Vogue and many other titles at Conde Nast over the years, and both have issued objections or denials to what has emerged.
"I believe strongly in the value of remorse and forgiveness, but I take the allegations very seriously, and we at Conde Nast have decided to put our working relationship with both photographers on hold for the foreseeable future."
Photo caption: Bruce Weber denies the claims