"My handshake with him, it's not innocent," Macron told the Journal du Dimanche in an interview published on Sunday.
"It's not the alpha and the omega of politics, but a moment of truth," CNN quoted the French President as saying in the interview.
The two newly-minted leaders met in Brussels on May 25 on the sidelines for the North Atlantic Treaty Organisation (NATO) Summit and shook hands in full view of the press.
In the interview, Macron compared his own handshake to his leadership posture.
"One must show that we won't make little concessions, even symbolic ones," Macron said.
The centrist European leader also likened Trump's diplomatic approach to those of Turkish President Recep Tayyip Erdogan and Russian President Vladimir Putin.
"Donald Trump, the President of Turkey or the President of Russia are of a mindset of power relations, which doesn't bother me," Macron said, adding: "I don't believe in diplomacy of the public invective but in bilateral dialogues. I don't let anything go. That's how one makes oneself respected."
Macron concluded the interview by saying that he believed he could establish a "cordial relationship" with Trump.
The tense moment during Trump's first trip abroad as President became the latest of his handshakes to draw attention online, reports CNN.
In February, Trump's lengthy handshake with Japanese Prime Minister Shinzo Abe went viral.