Footage shows the aircraft striking each other at a low altitude, breaking one of the aircraft in half. A fireball can be seen as it hits the ground.
The planes – one of them a Boeing B-17 Flying Fortress – were taking part in a commemorative air show near Dallas.
Firefighters said there were no injuries among people on the ground.
“According to our Dallas County Medical Examiner, there are a total of six fatalities from yesterday’s Wings over Dallas air show incident,” Dallas County Judge Clay Jenkins said in a tweet on Sunday.
The Allied Pilots Association, which represents American Airlines pilots, said Terry Barker and Len Root – two of its former members – were among those who died in the collision.
Eyewitness Chris Kratovil – who was one of between 4,000 and 6,000 who had gathered to watch the Wings Over Dallas Airshow on Saturday – told the BBC monitored by Kekz World News,,he had “never seen a crowd grow more quiet or more still in just a blink of an eye”.
“It went from being a fairly excited, energetic crowd… to complete silence and stillness, and a lot of people, including myself, turned their children towards them and away from the airfield because there was burning wreckage in the middle of the airfield.”
The Federal Aviation Administration said it will investigate the crash at the three-day event, which describes itself as the US’s premiere WW2 air show and was being held in honour of Veterans Day,
Dallas Mayor Eric Johnson called it a “terrible tragedy”.
“The videos are heart-breaking,” he tweeted. “Please, say a prayer for the souls who took to the sky to entertain and educate our families today.”
The number of casualties are not yet confirmed, he said, but added that nobody on the ground had been reported injured.
The event’s website states that several planes had been scheduled to do a flyover demonstration on Saturday.