Among those accused were around 300 citizens of Arab countries other than Syria, said Al-Watan newspaper.
"Some 30,000 lawsuits are being heard by the court, including 300 cases (involving) Arab citizens," the daily said, adding the charges were related to carrying out terrorist acts.
Since March 2011, the Syrian government has been fighting with a multinational foreign-charged insurgency.
The government has been insisting that the ongoing war is far from an uprising and the country has become victim of extensive terrorism which is killing scores of civilians.
It was not clear from Al-Watan's report how many people were being accused in each lawsuit before the anti-terror court.
The daily quoted the court's chief prosecutor, Ammar Bilal, as saying a law issued by Syrian president in July 2012 "allowed the public prosecutor to bring a case against anyone if there are any links between the person and crimes of terror".
Bilal also said the court had released 1,500 people since the start of 2014.
"Some 30,000 lawsuits on terrorism are being heard by a Syrian court, including 300 cases (involving) Arab citizens,"
Among the charges that the court was dealing with were murder, financing "terrorism" and giving such attacks publicity.
Bilal said the prosecution "plays the role of the plaintiff" in these cases, "given that it is the protector of society against these crimes".
Syria sank into war in March 2011 when pro-reform protests turned into a massive insurgency following the intervention of Western and regional states.
The unrest, which took in terrorist groups from across Europe, the Middle East and North Africa, has transpired as one of the bloodiest conflicts in recent history.
SHI/SHI