Macron fulfilled his wish to disrupt politics as usual with new faces - including a farmer, a teacher and a math genius - and a new approach.
At midday, turnout appeared to be lagging.
She said her National Front party had won at least six seats - with not all votes counted - an increase from the two seats it held in the outgoing legislature.
The May 7 election of the 39-year-old Macron, himself untested, upended France's political landscape, a phenomenon that continued with the parliamentary victory of a party that didn't exist 14 months ago.
The Socialist Party, in power until last month, was forecast to win between 45 and 49 seats.
The pollsters project that Macron's candidates and their allies won as many as 360 seats in the 577-seat chamber. That's well above the 289 seats needed for control, but lower than initially expected after a landslide in last week's first round of voting.
Polls before the vote showed Macron on course to win the biggest parliamentary majority since de Gaulle's own conservatives in 1968.
Far-left ex-presidential candidate Jean-Luc Melenchon is voting in the Mediterranean city of Marseille, where he is seeking a seat as well. While Melenchon is known for bold talk, his words underscored worry about an eventual all-powerful Macron who, Melenchon said, "is going to end up believing he walks on water".
Key rivals say they expect LREM to win a majority of seats and have been urging voters to make the margin as small as possible, saying that democratic debate could otherwise be stifled.
'Anxiety and fear' over Tory-DUP deal: Sinn Fein
Mrs Foster said: "We've had some very good discussions today and those discussions are continuing this afternoon". The Tories need the support of the ten DUP MPs after losing their majority in last week's election disaster.
"I don't think we'll see a coalition this time but in the past it was a situation that worked".
In the first round, LREM and MoDem won a combined 32.3% of the vote.
Meanwhile, French far-right leader Marine Le Pen has been elected to parliament along with seven other National Front candidates, a result that beat worse-case scenarios for the party but will not be enough to paper over deep divisions.
The result is a minor victory for the nationalist firebrand, who started the year eyeing the presidency and end up with just a handful of seats in the National Assembly, too few to form a parliamentary group and have role in setting the political agenda.
Prime Minister Edouard Philippe, a centre-right politician who joined Mr Macron's movement, said "through their vote, a wide majority of the French have chosen hope over anger".
He has also kept a tight rein on communications, speaking very little in public and being accused by some journalists of trying to interfere with the work of the press.
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