The group said it had received phone calls from relatives of prisoners asking Bashar al-Assad, Syria's president, to help bring an end to the violence.
There was no immediate comment from the Syrian authorities.
"Based on what my staff saw, the administration of the prison has cancelled visits today," Muhanad al-Hasni of the Syrian Human Rights Organisation, told Al Jazeera.
"We could see columns of smoke from inside the prison. We can't tell if it is gas bombs or fire. We were denied access into the prison."
The Saydnaya jail holds political prisoners including dissident intellectuals besides ordinary criminals.
Syria has recently faced criticism from the West for its crackdown against political dissidents.
Most of the arrests have been carried out under 45-year-old emergency laws.
At least 14 signatories of a December petition calling for democratic change in Syria are in dentention, including Riad Seif, a former member of parliament.
Saydnaya prison was built in 1987 and can hold up to 10,000 prisoners, according to the Syrian Human Rights Committee.
In 2004, the jail held several hundred members of the Muslim Brotherhood as well as left wing activists, Palestinians and detained Syrian army soldiers, the rights group says.