Henry is faced with the biggest decision of his post football career with The Daily Mail reporting that Wenger has told the club legend he can only keep his coaching role if gives away his media job and devotes himself full-time to the Arsenal Academy.
It is believed the Arsenal's record goalscorer is displeased with the ultimatum and could turn to coaching at other clubs.
The Frenchman has worked with various age groups in Arsenal's academy to qualify for his UEFA A licence.
He is currently completing his Pro Licence coaching badge and would clearly prefer to stay with his former club, where he had been offered the position of assistant manager of the Under-18s by head of the academy Andries Jonker.
Wenger is believed to have had issues with the former striker's television role ever since Henry claimed in April 2015 that Arsenal would struggle to win the Premier League with Olivier Giroud as their first-choice striker.
Henry's Sky Sports deal is allegedly worth $6.9 million per year.
Wenger was himself a television pundit for BeIN Sport during EURO 2016 and is rumoured to be one of the top candidates for the now-vacant job as manager of England's national team.