Even members of opposition political parties are welcome to come back, RIA Novosti reported Friday.
Returnees would receive "all-embracing assistance," the ministry said in a statement.
The U.N. High Commission on Refugees said as of Tuesday more than 678,000 refugees have crossed the Syrian border, with most of them traveling to Turkey, Lebanon and Jordan, CNN reported.
Each of those countries has now accepted more than 150,000 Syrians.
Some 3,581 Syrians crossed from Syria into Jordan from Wednesday afternoon to Thursday morning, said Anmar Hmoud, Jordan's government spokesman for Syrian refugee affairs.
That broke a one-day record that had only been set Sunday.
A U.S. delegation visiting the Islahiye Refugee Camp in Turkey found 8,825 people there.
Islahiye District Governor Osman Beyazyildiz said the camp, established in March 2012, has cost more than $17.8 million.
U.S. Ambassador to Syria Robert Ford noted Thursday that members of the regime have also left.
Anisa Makhlouf, President Bashar Assad's mother, is now in the United Arab Emirates, Ford said, while the president's sister is living in Dubai.
The Syrian Network for Human Rights, based in London, said 103 people died across Syria Thursday. The total included six children, nine women and 43 members of the Free Syrian Army.
One other person died under torture, SNHR said.