Police are investigating a complaint from the Vote Leave campaign that a Ryanair ad offering cheap flights to expats who want to vote to remain in the European Union breaches bribery laws.

The Ryanair “Brexit special” says overseas voters can “fly home to vote ‘remain’” from €19.99 on the day of the referendum, 23 June, or the day before, 22 June.

Vote Leave campaign director Dominic Cummings described the offer as “corrupt” and alleged it broke referendum rules and section 1 of the Bribery Act 2010.

Ryanair owner Michael O’Leary, a prominent backer of the remain campaign, said the complaint was “desperate” and in response extended the airline’s offer by a further 24 hours on Friday.

In a two-page complaint to Met police commissioner Sir Bernard Hogan-Howe, Cummings said: “By its discount on flights on the day before, and the day of the referendum, Ryanair is paying the expense, in part, of provision to voters in order to influence them to vote in the referendum.

“This appears to be corrupt, since the company is offering discounts on the commercial rate to customers with the sole aim of ensuring that they vote and vote to remain in the European Union.”

Scotland Yard confirmed it had received the correspondence and said it would consider its contents and respond in due course.

O’Leary, who this week appeared alongside chancellor, George Osborne, Vince Cable and Ed Balls, arguing that air fares would increase if the UK voted to leave the EU. said: “Vote Leave must be getting really desperate if they are now objecting to low fare air travel for British citizens.

“Ryanair’s Fly Home to Vote Remain seat sale ... fully complies with Ryanair’s policy of lowering the cost of air travel to/from the UK.

Cummings asked Hogan-Howe to “investigate this matter with the utmost expedition” and compared the offer to a pub next to a polling station offering cut-price drinks to people who voted a certain way.

He said the law should not allow the “democratic process to be gravely undermined by the wealth, power and influence of multinational companied which have a direct financial interest in preserving the corrupt regulatory system of the EU”.

About 5.5 million British citizens are estimated to live outside the UK, with at least 1.2 million of these living in other EU countries, but only a fraction are on the electoral roll.

Anyone who was registered in a UK constituency during the past 15 years is entitled to vote in British elections, but half of British expats are not aware of this fact.