Wenger has been in charge of Arsenal since 1996 but has not won the Premier League title since 2004.
He has come under enormous pressure in recent weeks and back-to-back defeats – to Watford and Chelsea – have brought the fans out in the droves.
Banners were displayed at Stamford Bridge last week and now Wright says Wenger looks ‘winded’ and thinks he could walk away in the summer.
“I was with the boss last night and if I’m going to be be totally honest I get the impression that that’s it,” Wright said on BBC 5 Live Sport.
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“I genuinely believe, I was with him for a few hours last night and we’re talking and obviously he didn’t say to me ‘I’m leaving at the end of the season’ but you get that feeling simply because I just get the impression in him and looking at him that’s it. I think that’s it.
“He actually mentioned while we were talking last night that it’s coming to the end – I’ve never heard him say that.”
When asked to described Wenger’s demeanour Wright said he looked like someone had taken all the wind out of his stomach.
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“He looked tired,” Wright added.
“I’ve got to say he looked really sharp to be honest and on different things that we were talking about fully of light you could see he’s switched on, just a wonderful articulate man and a lovely guy and you just feel he looks winded, like someone’s just whapped him in the stomach, he’s lost all his wind.
“I feel if someone’s going to ask me ‘do you think he’ll go at the end of the season?’ I would say I think he will.”