Although Macron publicly mocked Trump's stance on climate change and staged a much-publicised white-knuckle handshake with him at a Nato summit in May, he has invited the US leader and his wife Melania to France's national day celebrations in Paris on 14 July.
The official reason for the invitation, which Macron repeated during a phone call with Trump on Tuesday, is that this year the event will mark the 100th anniversary of the US joining the war with French troops in World War I.
Despite political differences, the US is France's principal military ally.
The US has yet to accept the invitation, French government spokesman Christophe Castaner said on Wednesday, but French officials are "working on the hypothesis that President Trump is coming".
Left-winger slams invitation
The invitation sparked an angry response from hard-left leader Jean-Luc Mélenchon.
"I deplore it," he told Europe 1 radio on Wednesday. "Mr Trump is not at all welcome at the 14 July celebration. The 14 July celebration is the celebration of the French people's liberty."
Trump is "a violent man" and represents the Nato military alliance, the France Unbowed leader said.
A Trump state visit to Britain appears to have been put on hold.
Following widespread anger at Prime Minister Theresa May's invitation, Trump reportedly told her he did not want to come if there would be large-scale protests against his presence in the country.
Syria warned over chemical attacks
Trump and Macron also agreed to stage a joint response if the Syrian government stages another chemical attack in the country's ongoing civil war.
On Monday the White House accused President Bashar al-Assad's government of preparing a potential chemical attack and warned it would pay a "heavy price" if it did so.
The statement prompted criticism from Assad allies Russia and Iran.
A Pentagon spokesman said US intelligence had noticed suspect activity at the launch site of an apparent chemical strike on a rebel-held town in April.
Shortly after that strike the US launched a cruise missile strike on the airfield in retaliation, the first direct US attack on the Syrian regime.
The French foreign ministry refused to say Tuesday whether it too had information on Syrian regime preparations for a chemical attack.
After a meeting last month with Russian President Vladimir Putin, Macron declared there was a "very clear red line" over the use of chemical weapons "by anybody" and warned of reprisals.
France is part of an international coalition that has been strking targets associated with the Islamic State armed group in Syria and Iraq since mid-2014.
The US has recently involved in confrontations with Assad's forces.