Campaign posters reading “Oui, on peut” – a translation of the former US president’s popular 2008 campaign slogan, “Yes, we can” – have appeared around Paris.
Mr Obama is not a French citizen and is thus ineligible to run.
But those behind the prank say the point is to highlight the lack of inspirational candidates.
Their message to the candidates, one of the organisers told the BBC, is: “Hey guys, you really don’t make us dream.”
French voters go to the polls on 23 April, and, unless one polls more than 50% in the first round, they will return on 7 May to decide between the two frontrunners.
A dominant figure in the campaign has been far-right leader Marine Le Pen, who has benefited from claims of financial wrongdoing which have swirled around the campaign of her centre-right rival Francois Fillon since late January, culminating last week in the announcement that a magistrate was launching a formal inquiry into the claims.
Current opinion polls give her a lead of several percentage points over Mr Fillon and centrist Emmanuel Macron – though both of her rivals are predicted to beat her in a two-way runoff.
It comes after two of the leading candidates in this year’s election race have been hit by a series of scandals.
The website, called Obama2017.fr, said:
“The French are ready to make radical choices. That is good because we have a radical idea to propose to them.
“Faced with the announced failure of the next presidential election, we think it is time to move to the 6th Republic to get France out of its lethargy.
“Barack Obama has completed his second term as President of the United States on January 21, why not hire him as President for France?”
Posters have been put up in Paris promoting the campaign and the organisers have been interviewed on several national TV stations.
Since then several US and other European stations have featured the people behind the stunt.
The website says it aims to collect one million signatures by 15 March in order to persuade the former US president to stand.
It may struggle however, reports BBC News.