Ryanair has called on Forsa to cancel its planned strike on July 12.
The budget airline claims the strike is supported by only 27% of its Irish pilots and has no mandate.
It has offered to meet the union early next week to discuss issues of seniority and base transfers.
"Ryanair's Irish passengers should not be disrupted next Thursday when this strike has the support of just 94 of Ryanair's over 350 Irish pilots.
"This is a quorum of just 27%, and significantly less than one-third of Ryanair's Irish pilots."
The airline claims the strike is being organised by Aer Lingus pilots, who will benefit from disruptions to Ryanair's schedules next week.