All the people taken to hospitals had minor injuries, according to a post on the social platform X by Denver International Airport.
Flight 1006, which was headed from the Colorado Springs Airport to Dallas Fort Worth, diverted to Denver and landed safely on Thursday afternoon after the crew reported engine vibrations, the Federal Aviation Administration said.
While taxiing to the gate, an engine on the Boeing 737-800 caught fire, the FAA said.
Photos and videos posted by news outlets showed passengers standing on a plane's wing as smoke surrounded the aircraft.
The FAA said passengers exited using the slides.
The airline said the flight experienced an engine-related issue after taxiing to the gate.
There was no immediate clarification on exactly when the plane caught fire.
The 172 passengers and six crew members were taken to the terminal, airline officials said.
"We thank our crew members, DEN team and first responders for their quick and decisive action with the safety of everyone on board and on the ground as the priority," American said.
Firefighters put out the blaze by the evening, an airport spokesperson told media outlets.
The FAA will investigate.
The country has seen a recent spate of aviation disasters and close calls stoking fears about air travel, though flying remains a very safe mode of transport.
Recent on-the-ground incidents have included a plane that crashed and flipped over upon landing in Toronto and a Japan Airlines plane that clipped a parked Delta plane while it was taxiing at the Seattle airport.
