Jardim took over as Monaco boss in 2014 and led the club to the Ligue 1 title in 2016-17, as well as reaching the semi-finals of the Champions League that year.
However, the club have made a poor start to the new season, failing to win any of their last 10 games since the opening day of the league campaign including losing each of their last four outings.
Jardim leaves Monaco with the team sitting 18th in the Ligue 1 table, only outside the automatic relegation places on goal difference.
"I am grateful and proud to have been able to coach AS Monaco for more than four years. I have always given the best of myself and worked with passion. We have won great victories together and I will always keep those memories in me," Jardim told the club's official website.
Vice-president Vadim Vasilyev added: "I want to greet with deepest respect Leonardo for all the work done. Leonardo has established himself on the bench of AS Monaco as a reference in Europe and leaves behind a very positive balance sheet.
"His passage will remain as one of the most beautiful pages in the history of the club. Leonardo will still be part of the AS Monaco family. "
Henry has been heavily linked with the post in recent days, reportedly turning down Aston Villa in favour of taking over at the Stade Louis II.