Jardim's position in Monaco has been under scrutiny since August after the club made a faulty start to the new campaign whilst bookmarkers expected them to be competing for the title with runaway leaders PSG.
He has recently been linked with jobs at Aston Villa and Bordeaux, but in both cases negotiations broke down after initial meetings.
However, should Monaco actually sack Jardim, then instead of handing their former striker his first job in the managerial arena The Monégasques should instead lure Arsene Wenger back for a second spell at the helm.
The 44-year-old Portuguese coach had to cope with the sale of numerous stars of his title-winning squad over the last two seasons.
But, after winning 3-1 at Nantes on the opening day of the season, Monaco failed to win any of their subsequent eight league games and lost to Atletico Madrid and Borussia Dortmund in Champions League Group A, which also includes Club Bruges.
Meanwhile, a summer spending spree headed by the signing of Russian World Cup star Aleksandr Golovin has not yet produced positive results for a club owned by Russian billionaire Dmitry Rybolovlev.
"I am grateful and proud to have been able to manage Monaco for over four years", Jardim said.
"His time here will remain as one of the most lovely pages in the history of the club".
Henry started his playing career at Monaco in 1994 and he was part of the side that won the French title in 1997 under Jean Tigana.
Born in Venezuela to Portuguese parents, he was appointed by Monaco in June 2014 and led them to third-place finishes in his first two seasons in charge.
If he does take over from Jardim, Henry's first game in charge could be the match against Lebo Mothiba's Strasbourg after the worldwide break.