The petition launched on Monday has already gained nearly 400 signatures with a current goal of 500.
“Due to the dramatic situation in which we find ourselves after the Brexit vote, we ask the Spanish government for an act of generosity for the British residents of Spain,” the petition reads.
“For many thousands of Brits in Spain and many thousands of Spaniards in the UK, the future is uncertain and worrisome. They are desperate.”
Journalist and author Giles Tremlett, who has written for The Guardian and The Economist, started the petition with fellow journalist William Chislett because of what dual citizenship would mean both in terms of his personal identity and for practical reasons, he told The Local.
"I feel that I have a dual identity after more than 20 years," Tremlett said.
Tremlett, currently based in Madrid, has lived in Spain for some 25 years and thus was unable to vote in the EU referendum due to the 15-year limit for Brits living abroad.
"With Brexit, I've effectively been expelled from my European citizenship against my choice and with no say in the matter."
The petition comes after Germany's Vice Chancellor said on Saturday that his country should consider offering dual citizenship to young Brits, and also encouraged other EU countries to do the same.
Rajoy has assured Brits living in Spain though that they are in no danger of losing their rights overnight.
He also points out that while Spaniards in Britain have the option to apply for dual citizenship after living there for five years, Britons in Spain do not have that option: To be granted Spanish citizenship, one must generally renounce their original nationality.
"My sons who were born and raised in Spain with British nationality do not have that right."
The petition also argues that a legislative change to grant such double citizenship is not without precedent. Last year, current acting Prime Minister Mariano Rajoy's government offered double citizenship to descendants of Jewish people forced out of Spain in 1492 during the Spanish Inquisition.
Tremlett says dual citizenship would solve many, though not all, of the problems Brits are currently grappling with.
"Spain has been a member of the EU for 30 years, Britain for 40 years," he said. "That's long enough for people to have based their entire life plans, their careers, their families, etc on being a European citizen.
"It's both traumatic and dramatic to have that snatched away."
Note: This article has been updated to add in the name of the petition's co-author.