Henry, who is currently working as Belgium assistant manager alongside Roberto Martinez, is the favourite to replace Jardim, whose Monaco side are languishing in 18th in Ligue 1.
With only one league win from nine games, the club have reportedly turned to Les Blues all-time record goal-scorer.
L'Equipe reports the former France international, who has also been linked with Championship side Aston Villa in recent days, is Monaco's first choice.
The former Arsenal star is understood to have impressed the Villa hierarchy in an interview but the lure of the side he started he professional career could sway his overall decision.
Monaco have not won since the opening day of the season in any competition and Sunday's loss at home to Rennes extended their consecutive defeats to four games.