Three survey firms - TNS Sofres, Opinion Way and Ipsos - concurred that the Socialists, the Ecologists and radical Left Front would win between 289 and 368 seats in the 577-member house next week. The UMP would get at least 270 seats, while the National Front and centrist MoDem party would each have one or two seats.
The National Assembly has far less powers than the Lower House in Anglo-Saxon democracies. France has an executive-style system in which huge authority is invested in the president.
Even so, a president is forced to compromise if the Assembly is dominated by a party of the opposing political hue. These periods of so-called cohabitation have happened several times in the last 30 years.
In his election campaign, Hollande put forward a programme of tax increases, job creation in the education sector and subsidies for companies that hire young people and retain old workers. In Europe, he has been leading the charge to weaken the pro-austerity campaign headed by German Chancellor Angela Merkel.
Turnout was less than 60 per cent, which is low by French standards, and reflected voter fatigue after a two-round presidential ballot.
Among those who did well was National Front leader, Marine Le Pen - daughter of firebrand founder Jean-Marie Le Pen - who picked up 47 per cent at Henin-Beaumont in the northern rustbelt, while far-left leader Jean-Luc Melenchon trailed a distant third.
Le Pen's success in the presidentials, where she picked up 18 per cent of the vote, stunned mainstream politics. Yesterday, her party was credited with an estimated 13-14 per cent of the total, more than triple its showing in 2007.
But because the legislature is chosen on a first-past-the-post rather than proportional basis, her party is likely to end up with only one or two seats, and possibly none at all.
Le Pen blasted the electoral system as a fraud that denied "millions of voters a voice".
She said: "Tonight, I confirm that we are the third biggest party in France."
Only a couple of dozen seats were determined outright in the first round, where popular candidates won at least 50 per cent of votes cast.
The runoff vote is held among any candidate that garnered 12.5 per cent or more in the first round, and the winner is the person who gets most votes.
The system usually makes it harder for fringe parties to win - but it also throws up intriguing scenarios whereby a left- or right-wing bloc of votes can be split, allowing a minority candidate to squeeze through.
A number of senior politicians are facing this threat.
Among them are former Socialist presidential candidate Segolene Royal - Hollande's ex and the mother of his children - and centrist leader Francois Bayrou. Both will have to fight hard to keep their seats.
