The former Hollywood producer and convicted sex offender is now in isolation, according to Michael Powers, president of the New York State Correctional Officers and Police Benevolent Association.
Weinstein, who was behind films including Shakespeare in Love and Pulp Fiction, is being held at Wende Correctional Facility, near Buffalo in upstate New York.
Mr Powers said several members of staff had been quarantined at the prison over the Covid-19 spread.
Weinstein, 68, was sentenced on March 11 to 23 years in prison . He was found guilty of committing a first-degree criminal sexual act against production assistant Miriam Haley in 2006 and of the third-degree rape of an aspiring actress in 2013.
New York jurors acquitted him of the most serious charges, of predatory sexual assault, which could have seen him given an even longer jail term.
His lawyers have vowed to appeal against his convictions.
Imran Ansari, a lawyer for Weinstein, said his legal team had not been informed of the coronavirus diagnosis.
"Given Mr Weinstein's state of health, we are of course concerned, if this is the case, and we are vigilantly monitoring the situation," he said.
Before arriving at Wende, Weinstein had spent time at Rikers Island, a prison in New York City and a hospital where he was treated for heart problems and chest pains .
He also faces separate sexual assault charges in Los Angeles.