European countries including France have been seeking to boost defence spending and increase weapons production in the face of possible US security disengagement and Russian aggression linked to the war in Ukraine.
Macron said on Tuesday that France was going to "increase and accelerate Rafale orders".
Dassault Aviation chief executive Eric Trappier said the company had increased output from one war plane a month in 2020 to more than two per month this year, and was working with suppliers to be able to produce combat planes even faster.
"We are planning to deliver three per month next year, and four from 2028-2029," he told Le Journal du Dimanche newspaper.
"We have heard the president's call and are studying the possibility of ramping up to five Rafale per month. There are no concrete orders yet, but we want to be ready," he said.
However, he did not say when this might be possible.
Trappier said that, if the French government approved, the company would also be "ready to provide its services" to any country reviewing its orders for US-made F-35 combat planes since President Donald Trump took office.
Germany on Friday said it was committed to buying F-35 fighter jets despite reports that it was reconsidering due to worries about an over-reliance on US defence supplies.
But Canada said last week it was reviewing a major purchase of F-35s amid serious tensions over tariffs and Trump threatening to annex the country.
That announcement came two days after Portugal said it, too, was re-examining a possible purchase of F-35 fighter jets.
Trappier said that Portugal had not yet reached out to his company.
Last year, France's air force had 108 Rafale jets, and the navy had 41. France was due to receive 56 additional aircraft before Macron's announcement.
The defence minister last month said the air force needed 20 to 30 more Rafales to face a crisis scenario.
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