Staff are understood to have raised concerns over the pilots’ behaviour before the Air Transat A310 took off carrying up to 250 passengers on board.
The Canadian airline’s plane was due to depart from Glasgow to Toronto on Monday afternoon, The Press Association reported.
Officers arrested the men on suspicion of being “impaired through alcohol” before they were due to take off at 1pm.
A spokeswoman said: “Police Scotland can confirm that two men aged 39 and 37 years have been arrested in relation to being allegedly impaired through alcohol under the Railway and Transport Safety Act 2003 (Section 93).”
The Canadian airline confirmed the arrest of its employees, adding that the flight had been rescheduled and the passengers put up in hotels.
A spokesman said: “Air Transat has learned of the arrest on July 18 of two crew members assigned to its Glasgow-Toronto flight.
“We will await the results of the investigation and judicial proceedings before commenting on the matter before making any further comments. The safety of our crews and passengers is, and will remain, a top priority at Air Transat.”
Some passengers voiced their anger over an apparent lack of an official explanation over the delay.
Toronto resident Nick Davis tweeted that he had been “Ignored all day by @airtransat”, while @sdp0313 tweeted: “TS725 22 hr delay. Thinking the op issue was more of a staff issue, police involved, why? Answers plz @airtransat.”
Apologising for the delay to Sean Brown, who had been on holiday in Scotland from Ontario, the airline tweeted “for operational reasons we have no choice”.
A rescheduled flight is expected to depart on Tuesday.