Royal expert Richard Fitzwilliams believes Macron’s move was reflective of his “forceful” personality.
He told The Express: "When President Biden visited King Charles in Windsor in July there was much comment when he put his hand on the King’s back. It was obviously meant to be friendly.
"Today we see President Macron extend extremely warm body language to King Charles. In 2009, Michelle Obama, in London during President Obama’s first visit to the UK, put her arm around Queen Elizabeth’s shoulder.
"The Queen responded by putting her arm around the First Lady’s waist. She quoted the Queen as telling her that royal protocol is 'rubbish'.
Macron has a close relationship with King Charles
"It’s fascinating to see how different world leaders respond to each other. Macron has a forceful personality and the way he greeted the King was an expression of it. It was meant to show warmth."
It’s not the first time royal protocol has been broken around King Charles, after US President Joe Biden was seen touching the monarch in July.
Charles and Camilla are visiting Paris and Bordeaux as part of the state visit.
They attended a ceremony of remembrance and wreath-laying at the Arc de Triomphe in the centre of the capital.
Charles symbolically lit the monument’s eternal flame, which burns in memory of those who died in the First and Second World Wars.
It was the first time in 30 years the ceremony has been included in a state visit.
Charles and Camilla were seen exchanging warm words with Mr and Mrs Macron throughout the ceremony.
The French and British national anthems were played and there was a flypast by the Patrouille de France and Red Arrows before the couples travelled down the Champs Elysees by car.
Charles will become the first British monarch to give a speech from France’s senate chamber on Thursday.