The Syrian Observatory for Human Rights said the aircraft, believed to be from an Arab state, went down near Raqqa.
The UK-based group said the fate of the pilot was unknown, but IS supporters published photos they said showed a Jordanian airman surrounded by gunmen.
If confirmed, this would be the first coalition warplane shot down since air strikes on IS began in September.
The authorities in Jordan, one of four Arab countries whose aircraft have been flying sorties over Syria, has so far not commented on the incident.
The Syrian Observatory said it had received reports from its network of activists that IS members had taken “an Arab pilot prisoner after shooting his plane down with an anti-aircraft missile near the city of Raqqa”.
Earlier, the pro-IS Raqqa Media Center had posted photos on its Facebook page purportedly showing armed men holding the captured pilot, with a caption identifying him as Royal Jordanian Air Force flight lieutenant and giving his name.
The man appeared able to stand but was bleeding from the mouth. He was wearing only a T-shirt and underwear.
It was not possible to verify the images.
The jihadist group claimed to have shot down the coalition warplane with a heat-seeking missile.
The air forces of Jordan, the US, UAE, Saudi Arabia, and Bahrain have carried out hundreds of air strikes on IS in Syria in the past three months.
Many of the targets have been in and around Raqqa, which is the de facto capital of the “caliphate” whose creation IS proclaimed in June.
Syrian government warplanes also regularly bomb Raqqa and the surrounding province. On Tuesday, an air strike killed more than 20 people, according to the Syrian Observatory.
Australia, Belgium, Canada, Denmark, France, the Netherlands and UK have joined the US in conducting air strikes on IS in neighbouring Iraq.
The reports will raise concern among the coalition nations about the level of armament available to the militants and the defensive measures deployed by coalition jets.
It may further diminish the appetite of Arab nations to take part in such operation, our correspondent adds.