Aer Lingus said it contacted Ryanair to inquire if seats were available during yesterday's strike but was told its flights were full.
Ryanair said it was not contacted by Aer Lingus and still had seats available yesterday.
"It is factually incorrect and absolute rubbish . . . they definitely did not contact the airline," a spokeswoman for Ryanair said.
"The company has had capacity throughout the day," she said. Ryanair was unable to state that capacity. Aer Lingus said it contacted Ryanair twice.
"We rang Ryanair yesterday [Monday] and the message was all flights were full. We rang again this morning just to check and were told there were about 20 seats available and on a couple of flights they were full," Mr Declan Conroy, of Aer Lingus, said.
"Today we heard they had lots of seats," he said.
According to British Midland, about 40 passengers were booked onto its flights to London yesterday when no seats were available. They said this could have been done by Aer Lingus staff or travel agents.
British Midland said Aer Lingus were directing passengers towards them regarding overbooked flights. "We've not been able to get on with normal duties," Mr Colin Cook, a member of British Midland's ground staff said.
A couple of thousand Aer Lingus passengers got flights with other airlines yesterday, according to Mr Conroy. He said most business people scheduled to fly with Aer Lingus cancelled their flights.
Aer Lingus offices in other countries contacted passengers flying to the Republic yesterday to inform them of the cancellations, Mr Conroy added.
Many of the 20,000 passengers affected by yesterday's cancellations rebooked onto other Aer Lingus flights later this week, he said.
"It was a dark day for Aer Lingus customers. But in the context of what we were faced with it worked out well," Mr Conroy said.