Video footage appeared to show British holidaymakers were involved in the brawl after the Boeing 737 took off from Amsterdam on Friday.
KLM said both men were restrained with the help of other passengers and were later arrested by Spanish police on arrival in Ibiza.
"Two unruly passengers refused to wear their face masks, and they were bothering their fellow passengers physically and verbally,” a spokesperson for the airline said.
"The pilot informed the local authorities and upon arrival, both passengers were arrested. The flight safety was not compromised during the flight."
The Instagram account which first posted the video on Sunday morning claimed one of the passengers involved was English and had been drinking vodka.
"Panic and violent brawl!," wrote @michighclub. "Unruly English passenger on board KLM flight to Ibiza, he had been drinking Gregus vodka.
"They refused to wear facemasks. 2 arrests were made."
Passengers were heard to shout: "Stop now, there are kids over here!"
Several other people in the video also appeared not to be wearing face coverings.
Masks are mandatory on KLM flights and at airports in the Netherlands as part of the country's coronavirus restrictions.
KLM said passengers were checked for masks on boarding and said staff also issued reminders during the flight.
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