Arsene Wenger still has the full backing of Arsenal's board but is growing increasingly concerned that the unrest among supporters could affect his players, according to British newspaper reports.
With Wenger under increasing pressure after Arsenal's FA Cup loss to Watford on Sunday, the Daily Mirror and Daily Telegraph reported that the Arsenal manager will be allowed to decide his own future as he heads into the final year of his contract next season.
But, citing sources close to Wenger, they said he may decide to step down if the negative atmosphere around the Emirates gets even worse during the last part of the season and starts affecting the team.
Arsenal's majority owner Stan Kroenke praised Wenger's analytical mind as recently as Friday at a sports analytics conference in Boston, and few would expect the Frenchman to be fired after his 20th season in charge.
But Arsenal's faltering Premier League title challenge and the likelihood of another Champions League exit in the round of 16 when they face Barcelona on Wednesday has caused an increasing divide among Arsenal supporters in recent weeks.
A banner urging Wenger to leave was held up by some traveling Arsenal fans after the 4-0 FA Cup win over Hull last Tuesday -- the Gunners' only victory in their last seven games -- and there was even a scuffle among some supporters outside the Emirates after the Watford loss.
Wenger, who was set to address media at the Camp Nou on Tuesday evening, has been given the backing of several former players over the last week, including Ian Wright and Patrick Vieira.
"Fans can be frustrated, but fans sometimes forget what Arsene has brought to this football club and I think he deserves a bit more respect," Vieira told TalkSPORT on Monday.
"Fans can boo and show their anger, but the way they did it with that banner showed too much lack of respect."
Wenger has not won the Premier League since 2004, but ended a nine-year trophy drought with back-to-back FA Cup victories in the last two seasons.
Arsenal are also the only club to play in the Champions League in all 19 of his seasons in charge, a feat that has secured the club's financial stability even as it spent years paying off the construction of the Emirates Stadium.
Their recent lack of form means another top-four finish could be under threat, though. Arsenal are only three points ahead of fifth-place West Ham, while trailing leaders Leicester by 11 points with a game in hand.
With star duo Mesut Ozil and Alexis Sanchez both having two years left on their contract, such a failure could dissuade them from extending those deals in the summer.
Wenger has been adamant that his team wants "to make the impossible possible" against Barcelona, and defender Per Mertesacker also said the team remains united in the midst of the turmoil.
"The season is not over. We play in a confident way, we maybe lack that bit at the end. The desire to play together and to be good as a team is there, you can feel that," the German said.