Antony joined United from Ajax in August 2022
The 23-year-old was due back to training on Monday after being dropped from Brazil's squad for their World Cup qualifiers against Bolivia and Peru.
But United say it has been agreed with the player that he will delay his return until further notice in order to address the matter.
Antony denies all the allegations.
"Manchester United acknowledges the allegations made against Antony," said a club statement.
"Players who have not participated in international matches are due back in training on Monday.
"However, it has been agreed with Antony that he will delay his return until further notice in order to address the allegations.
"As a club we condemn acts of violence and abuse. We recognise the importance of safeguarding all those involved in this situation, and acknowledge the impact these allegations have on survivors of abuse."
A police investigation has been opened following allegations of domestic abuse against his former girlfriend.
He is also facing further allegations of assault made by a woman in an interview with a Brazilian TV channel.
Antony appeared on Brazilian TV on Friday, saying "I never attacked any woman" and added that "the truth will come out".
He issued a statement on Sunday saying: "I have agreed with Manchester United to take a period of absence while I address the allegations made against me.
"This was a mutual decision to avoid distraction to my teammates and unnecessary controversy for the club.
"I want to reiterate my innocence of the things I have been accused of, and I will fully co-operate with the police to help them reach the truth. I look forward to returning to play as soon as possible."
Antony has been accused of attacking his former girlfriend Gabriela Cavallin "with a headbutt" in a Manchester hotel room on 15 January, leaving her with a cut head that needed treatment by a doctor.
She alleges she was also punched in the chest, causing damage to a silicone breast implant, which required corrective surgery.
Antony previously said on social media about Cavallin's claims: "I can calmly state that the accusations are false and that the evidence already produced and the other evidence that will be produced demonstrate that I am innocent of the accusations made."
Ingrid Lana, a 33-year-old banker, has also claimed in a short clip released by Brazil's Record TV that she hit her head when "pushed against a wall" by Antony at his house in Manchester in October 2022.
Both Greater Manchester Police and Sao Paulo Police are looking into the claims.
Antony did not play in the international break after the Brazilian football federation said he had been withdrawn from their squad after "facts became public" that "need to be investigated".
-- Courtesy of BBC Sport