The pilot reported a drop in cabin pressure soon after the jet took off from the southern Spanish city of Jerez with a family of three aboard and headed for Cologne in Germany, the Bild tabloid said.
The LETA Latvian news agency said that there were six people on board the plane.
The Austrian-registered plane lost contact with air traffic controllers while it was flying off the Iberian coast. European fighter jets took turns to shadow the "ghost plane" that reportedly showed no signs of people on board, according to Bild.
A Danish F-16 was scrambled to follow the jet which it reported seeing spin and crash into the sea as it was heading toward the Latvian coast at around 7:45 p.m. The Swedish coast guard sent rescuers to the crash area.
According to German media reports, the pilot may have lost consciousness prior to the crash.