He was attacked on stage, and is now on a ventilator and unable to speak, Andrew Wylie said in a statement, adding that the author, 75, may lose one eye.
Mr Rushdie has suffered years of Islamist death threats after writing The Satanic Verses, published in 1988.
Police detained a suspect named as Hadi Matar, 24, from Fairview, New Jersey.
New York State Police said the suspect ran onto the stage and attacked Mr Rushdie and an interviewer at the Chautauqua Institution in western New York state.
Mr Rushdie was stabbed at least once in the neck and in the abdomen, authorities said. He was taken to a hospital in Erie, Pennsylvania, by helicopter.
"Salman will likely lose one eye; the nerves in his arm were severed; and his liver was stabbed and damaged," his agent said.
No motive or charges have yet been confirmed by police, who are in the process of obtaining search warrants to examine a backpack and electronic devices found at the centre.
Police told a press conference that staff and audience members had rushed the attacker and took him to the ground, and he was then arrested. A doctor in the audience gave Mr Rushdie first aid.
The interviewer who was with Mr Rushdie, Henry Reese, suffered a minor head injury and was taken to a local hospital. Mr Reese is the co-founder of a non-profit organisation that provides sanctuary to writers exiled under threat of persecution.
Linda Abrams, an onlooker from the city of Buffalo, told The New York Times that the assailant kept trying to attack Mr Rushdie after he was restrained.
"It took like five men to pull him away and he was still stabbing," Ms Abrams said. "He was just furious, furious. Like intensely strong and just fast."
Salman Rushdie: The writer who emerged from hiding
Indian-born novelist Mr Rushdie catapulted to fame with Midnight's Children in 1981, which went on to sell over one million copies in the UK alone.
But his fourth book, published in 1988 - The Satanic Verses - forced him into hiding for nearly 10 years.
The surrealist, post-modern novel sparked outrage among some Muslims, who considered its content to be blasphemous - insulting to a religion or god - and was banned in some countries.
Several people were killed in anti-Rushdie riots in India and in Iran the British embassy in the capital, Tehran, was stoned.
In 1991 a Japanese translator of the book was stabbed to death, while a few months later, an Italian translator was also stabbed and the book's Norwegian publisher, William Nygaard, was shot - but both survived.
A year after the book's release, Iran's Supreme Leader Ayatollah Khomeini called for Mr Rushdie's execution. He offered a $3m (£2.5m) reward in a fatwa - a legal decree issued by an Islamic religious leader.
The bounty over Mr Rushdie's head remains active, and although Iran's government has distanced itself from Khomeini's decree, a quasi-official Iranian religious foundation added a further $500,000 to the reward in 2012.
There has been no reaction from the Iranian government to Mr Rushdie's stabbing. Iranian media were describing Mr Rushdie as an apostate - someone who has abandoned or denied his faith - in their coverage.
Salman Rushdie has faced death threats for more than 30 years since the publication of The Satanic Verses. Mr Rushdie said the main thrust of his novel was to examine the immigrant experience, but some Muslims were offended by portrayals of the Prophet Muhammad and the questioning of the nature of the revelation of the Quran as the word of God.
The Satanic Verses was banned first in the author's country of birth, India, and then several other countries before Iran's Ayatollah Khomeini issued his infamous fatwa.
The fatwa called for the killing of anyone involved in the publication of the book and offered rewards to those who took part in the murders. That fatwa has never formally been rescinded.
Surprised by the widespread nature of the protests, Salman Rushdie apologised to Muslims but went into hiding.
When Mr Rushdie was knighted in 2007 by the Queen, it sparked protests in Iran and Pakistan, where one cabinet minister said the honour "justifies suicide attacks".
Several literary events attended by Mr Rushdie have been subject to threats and boycotts - but he continues to write. His next novel, Victory City, is due to be published in February 2023.
Fellow authors such as JK Rowling and Stephen King have written messages of support.
Booker-prize winning author, Ian McEwan, called it an "appalling attack" that "represents an assault on freedom of thought and speech"
"Salman has been an inspirational defender of persecuted writers and journalists across the world. He is a fiery and generous spirit, a man of immense talent and courage and he will not be deterred," he added.
Writer Taslima Nasreen, who was forced to flee her home in Bangladesh after a court said her novel Lajja offended Muslim's religious faith, said she now feared for her own safety in the wake of Mr Rushdie's attack.
Source: BBC, Sam Cabral in Washington & Matt Murphy in London, August 13, 2022
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"One is absolutely sickened, not by the crimes that the wicked have committed,
but by the punishments that the good have inflicted." -- Oscar Wilde
but by the punishments that the good have inflicted." -- Oscar Wilde