A presidential source later said France could increase its military contribution to the USA -led coalition which - alongside the SDF - is fighting Islamic State in Syria.
"He assured the SDF of France's support for the stabilization of the security zone in the northeast of Syria, within the framework of an inclusive and balanced governance, to prevent any resurgence of Islamic State".
Mr Macron's offer came after he met a delegation of Syrian and Arab fighters in Paris on Thursday.
He warned that while Turkey had no intention of harming the troops of its North Atlantic Treaty Organisation allies deployed in Syria, he would not allow their presence to serve as protection for "terrorists".
The United States has 2,000 troops in SDF-held territory, and France also has some troops there as part of the coalition.
Turkey on January 20 launched Operation Olive Branch to remove PYD/PKK and Daesh terrorists from Afrin.More news: Ibrahimovic leads Galaxy rally past LAFC in MLS debut
More news: Declan Donnelly prepares to host without Ant
More news: Astronomers perplexed to find that distant galaxy has no dark matter
Macron's comments were not well received by Turkish President Recep Tayyip Erdogan, who said he was "extremely saddened by France's. wrong stance". Turkish forces are now conducting a military operation in north-western Syria, targeting Kurdish militias while Turkish officials repeatedly threatened to move further east along the border. The U.S. government is now sending very carefully calibrated signals to Turkey that U.S. forces in Manbij are not looking for a fight, but they are not leaving, either.
"Turkey's position on PKK/PYD/YPG, which seeks to legitimise itself as SDF, is perfectly clear", Kalin said, warning Paris against taking any steps which could be construed as legitimising terror groups. "Let the other people take care of it now", President Donald Trump said in a speech on Thursday.
"We have no need for mediation", Erdogan said.
Before the statement, Bozdag warned that those working with terror groups would become Turkey's target and hoped France would not take such an "irrational step".
The French presidency did not confirm that Macron had pledged more troops, but the presidential source said France could bolster its military intervention in Syria "within the existing framework" of the US -led coalition.