The petition's organiser James O'Malley, said the capital was "a world city" which should "remain at the heart of Europe".
Nearly 60% of people in the capital backed the Remain campaign, in stark contrast to most of the country.
The LSE's director said the vote showed how "radically different" London is.
Prof Tony Travers said London's economy and politics "look so different" to the rest of the country and it was up to the mayor to decide whether to argue for more power.
"Maybe moving more decision making to cities and councils could be a solution to the differences within the country," he said.
Following the result, Sadiq Khan said it was "crucial that London has a voice at the table during those renegotiations" with the EU.
"We will continue to look outwards and trade and engage with the entire world, including the European Union," he said.
The petition, which suggests the mayor could become "President Sadiq", has already been signed by more than 27,000 people.
Mr O'Malley said he was a "big EU fan" and was "fed up watching the results" when he set up the page on change.org.
He said he was "astonished" the petition had taken off but suggested it showed he had "clearly touched a nerve" with others who "like me want to live in an international city".
One person who commented on the page said he felt "morally, culturally and historically closer to Paris, Brussels and Rome than I do to Sunderland".
Another wrote: "We need to break free of the dead weight."
A second petition calling for London to remain part of the EU has been signed by more 7,500 people.
A petition calling for Sadiq Khan to declare London an independent state after the UK voted to quit the EU has been signed by thousands of people.