Two leading groups, Jaysh al-Islam and Ahrar al-Sham, which formed a pact last year, both say the plane was shot down.
Syrian state media said the crash was caused by a technical fault.
Jaysh al-Islam, the larger of the two groups, posted footage online which it claimed showed the pilot being held after ejecting from the jet.
The video, bearing Jaysh al-Islam's logo, showed an object engulfed in flames followed by an interview with the supposed pilot.
The man gave his name as Nawras Hassan, as well as his date of birth and hometown. He said he was based in Dumair air base and that he had been a pilot for 16 years.
A later image posted by the group appeared to show the pilot dead.
Jaysh al-Islam said in a statement that the pilot was captured by the Joint Operations Room - the name adopted by a collection of rebel groups in the region including Jaysh al-Islam, Ahrar al-Sham, and Jabhat al-Nusra.
Jaysh al-Islam said the pilot was captured by militants from the al-Qaeda-linked Jabhat al-Nusra, who were supposed to hand him over but killed him instead.
Neither the claims nor the footage have been independently verified.
More than 280,000 have been killed and millions displaced by Syria's five-year conflict, according to UN estimates.

The bird along with its dead mate was discovered at Wemyss Estate off Standing Stane Road in Kirkcaldy on Saturday 19 March.
The swan had been shot through the wing with an arrow which narrowly missed its vital organs and spine.
It underwent emergency surgery and was on antibiotics and painkillers until its release on 13 April.
The swan was cared for at the Scottish SPCA's national wildlife rescue centre in Fishcross and was released at a protected site on the River Tay.
Manager Colin Seddon said: "Once her wounds were healing and she was feeding well she went on to a wet pool and once given the all clear by our vet she went outside into one of our swan paddocks where she was able to join another recuperating swan.
"Being a flock animal, swans do much better when kept with other swans.
"After a week outside and once the weather conditions were suitable, she was released along with her new companion at a privately owned nature reserve on the River Tay, where the wild swans are support fed until they choose to disperse and go their own way."
Insp Sarah Gregory, of the Scottish SPCA, said: "Sadly the swan's mate was found dead at the same time but we have recently received the results of a post-mortem which do not show anything linking the death to intentional trauma or a bow and arrow attack.
"Despite our public appeal following the cruel and mindless attack on the female swan, we have not yet traced the perpetrator and we would still urge anyone with information to come forward."

The League One side have been under the embargo since November because of a £300,000 bill with HM Revenue & Customs and fees owed to other clubs.
The Latics are bottom of the table, three points from safety, and are the lowest scorers in the Football League.
"As far as I'm led to believe it should be cleared at the latest next week," Robinson told BBC Radio Manchester.
"There is money coming in from the league so hopefully we can clear that and we can try and crack on with things."
Oldham have not won a league game since 18 October in what is Robinson's first season in charge after arriving last summer.
"We've got plans," the 42-year-old added. "We've had plans since November or December time and we've spoken to a lot of players
"Obviously it is not an ideal situation as we've not been able to press the button on things. When we get the green light to go and sign people we'll see if they're still available."
Syrian rebels say they shot down a Syrian air force jet and blamed an al-Qaeda-linked group for killing the pilot.
[NEXT_CONCEPT]
A swan that was shot with an arrow in Fife has made a full recovery and has been released.
[NEXT_CONCEPT]
Oldham Athletic should be able to exit their transfer embargo within a week, says manager Stephen Robinson.