Wenger met Arsenal majority shareholder Stan Kroenke and chief executive Ivan Gazidis on Monday to discuss his future before his contract expires next month.
The 67-year-old is also due to attend an Arsenal board meeting on Tuesday when the directors will be told whether he will extend his 21-year reign.
Wenger has been heavily criticised this season, with Arsenal fans calling for his resignation after their team failed to qualify for the Champions League for the first time in 20 years.
Reports suggest Wenger is leaning towards signing a new two-year deal despite the abuse and talk of his frustration at boardroom plans to modernise his coaching staff.
With Arsenal having beaten Premier League champions Chelsea 2-1 in the FA Cup final on Saturday, giving Wenger a record seventh success in the competition, it seems unlikely the Frenchman will quit.
But if he does walk out, Wright, Arsenal’s second-highest goalscorer of all time, says it has been total incompetence from the club’s hierarchy to have allowed this situation to have arisen.
Writing in his column in The Sun, Wright said: “I’ve got a nasty feeling Arsene Wenger will walk away from the Arsenal job today, leaving the club totally in the lurch.
“And you would have to say it would serve them right, because there has been absolutely no forward planning for the day when he finally left the Emirates.
“The only way the board should have let him run down his contract is if they had private assurances he was staying.
“Otherwise, it is total incompetence to get to this stage.”