Seeking "revenge," Russian forces stormed a Syrian town on the border with Turkey hunting for the attacker who claimed he shot and killed the pilot of a downed Russian jet, state media reported Monday.

A Turkish F-16 fighter jet shot down the Russian aircraft near the Turkish border north of Latakia, Syria on Nov. 24, a senior U.S. official tells Fox News. Soon afterwards, Turkish citizen Alparslan Çelik said in a video that he killed the pilot on the ground because Russia had been dropping bombs minutes before the jet was downed.

"There is no place for a person who has bombed civilian Turkmens every day," Çelik said, according to Turkey's Hurriyet Daily News. He reportedly added, "Reprisal is the most natural right."

Russia claims special forces rescued the other man on the jet, a navigator.

State media report the gunman is in the town of Rabia, in Latakia province, which Syrian government forces recently recaptured from rebel fighters.

Russian president Vladimir Putin's government did not make clear what would happen to Çelik if soldiers found him, but they were looking to "take revenge," according to state media. Previously, Foreign Ministry spokeswoman Maria Zakharova ordered Turkey to "take immediate steps to apprehend Alparslan Çelik and his accomplices and bring them to justice for the murder of the Russian pilot."

The nearly five-year Syrian conflict that began in 2011 with protests against President Bashar al-Assad's rule, has morphed into an all-out war that has killed a quarter-million and displaced millions.

Fox News' Jennifer Griffin and The Associated Press contributed to this report.