The Socialists appear tied with the right-wing UMP party on about 35% of the vote, but the support of Green allies gives them closer to 40%.
The outcome of the polls is expected to determine the extent and pace of reform under the newly-elected French leader.
Run-offs are to be held a week later.
The early indications are that turnout has been much lower than in the presidential elections in April, at about 60%.
France’s 46 million eligible voters are picking representatives for 577 seats in the National Assembly.
TNS Sofres, Ipsos and OpinonWay pollsters all agreed that the political left, including the Communist-backed Left Front, would win at least 289 seats in the 577-seat Assembly and possibly as many as 368 seats.
These are predictions rather than hard results and it is hard to predict accurately what the final tallies will be before next week’s decisive round of voting.
But with the Senate already under the control of the Socialists, it appears that Francois Hollande will have a majority in the lower house – even if only with the support of allies – which would give him unprecedented power to force through his reform programme.
Francois Hollande’s government is due to present a revised budget plan to parliament next month.
The result of the parliamentary election will determine the pace of reform and how radical it becomes.
“It’s a good result tonight… but we have to remain mobilized for the second round,” Foreign Minister Laurent Fabius, an influential Socialist, was quoted by news agency AP as saying.
The election also saw a surge in support for Marine Le Pen’s far right National Front, which won almost 14% of votes, according to the exit polls – well above the 4% it got in the 2007 parliamentary elections.
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