Swissport is trying to avert a strike at Liverpool airport over the August Bank Holiday, which would involve around 200 staff.
The members of the GMB union have voted to walk out over health and safety issues, pay and working practices.
However, Swissport said it had put a new offer forward, which is now being considered by members.
In a statement the company said "We are aware that GMB has named three dates for potential industrial action. It’s important to note that the strike ballot was conducted on the basis of a previous offer and we are now in the final stages of proposing a revised offer for consideration.
"Negotiations are ongoing and we remain hopeful of reaching a resolution. Swissport remain committed to resolving this matter to protect the interests of our customers, airline passengers and employees."
If the strikes go ahead, they will be on August 22, 24 and 29.
