Some 80 percent of the French want him to dismiss Prime Minister Jean-Marc Ayrault, according to a Harris Interactive poll last week, and ambitious and tough-talking Interior Minister Manuel Valls comes top of their favourites for the job.
"I can't see how (Ayrault) could stay, unless we save about 30 towns, something nobody really believes," a Socialist Party source said.
(Read more: These women battle it out for Paris)
Dissatisfaction with Hollande's rule and a string of legal intrigues involving opposition conservatives are seen hitting turnout.
The streets of Paris were quiet on Sunday morning near the old stock exchange building in central Paris, without the queues at polling stations typically seen on an election day.
"I'm scandalized by these imbeciles who don't vote," said Yann Dedet, a 67-year-old cinema producer, told Reuters after casting his ballot "for the Left, of course".
Turnout in last weekend's first round was 63.5 percent - considered low by commentators in a country with a strong attachment to its mayors, who wield considerable local power.
A high abstention rate is seen helping the anti-immigrant FN win in the depressed post-industrial north and in southern towns such as Beziers and Avignon.
National Front breakthrough
Pollsters identify half a dozen FN-run towns emerging after the vote, giving the party a chance to try exercising power once more after its attempts to run four southern towns it won in 1995 and 1997 revealed its lack of competence.
The FN's made a striking breakthrough last Sunday by winning power outright in the northern town of Henin-Beaumont, a former coal-mining center with unemployment now around 18 percent.
Polls show the Socialists are favorite to hang on to Paris, where the gaffe-prone efforts of the conservative candidate to lure so-called "bobo" (bourgeois-bohemian) voters have been widely derided on social media.
The UMP's Nathalie Kosciusko-Morizet topped the popular vote in the first round, but Parisians vote for separate lists in each of the city's 20 arrondissements and the Socialists look set to have a majority of councillors who will pick the next mayor.
(Read more: Hollande braces for losses in French local vote)
In Marseille, right-wing incumbent Jean-Claude Gaudin, who has transformed France's second city with new seafront museums, a tramway and a pedestrianised old harbour area, looked set to win a fourth term after his Socialist rival came in third in the first round, trailing the FN candidate.
Despite the election losses, Hollande's government has said it will stick with economic reforms and spending cuts, including a plan to phase out 30 billion euros ($41 billion) in payroll tax on companies in exchange for hiring more workers.
A government source said Paris was also preparing tax breaks for households, which would raise new questions over whether France can fulfil a promise of bringing its public deficit down below the European Union target of 3 percent of gross domestic product.
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