The Portuguese's claim is the latest in a war of words between the two, whose respective sides are challenging for the Premier League title this season.
Earlier on Friday, Wenger declared that the league was Chelsea's to lose, adding that Mourinho's claim that the Blues are outsiders represents a fear of failure.
At a press conference later in the day, Mourinho - who won five major trophies during his first spell as Chelsea boss, including the Premier League twice, between 2004 and 2007 - drew attention to Arsenal's current eight-year spell without a major trophy, saying: "He is a specialist in failure, I am not.
"If he is right and I have a fear of failure it is because I don't fail many times. So maybe he is right?
"The reality is he's a specialist because eight years without a piece of silverware, that is failure. If I did that in Chelsea I would leave London and not come back."
When asked about Wenger's claim that Chelsea are favourites for the title, Mourinho replied: "The champions are Manchester United. They are the title-owner until the moment where mathematically they lose it.
"If they are not champions this season, and we all agree it is almost impossible for them to be, they lose the title, nobody else."
League leaders Chelsea hold a one-point advantage over second-placed Arsenal in the race for the title.
Manchester City are also in the running, two points worse off than the Gunners in third but having played a game less, with Liverpool fourth a point behind Manuel Pellegrini's side.
At his own press conference ahead of Arsenal's FA Cup fifth-round tie with Liverpool this weekend, Wenger was asked why, apart from himself and Pellegrini, every other manager is ruling out title success.
The Frenchman responded: "It's a fear to fail.
"If you're not in the race you cannot lose it, if you declare yourself not in the race you cannot lose it, simple as that.
"I think just that our job is to be ambitious and to try to win, if we do not win to take full responsibility for that. That's as simple as that and that's how I see it.
"I say yes we are in it, yes we will have a go for it, we will absolutely give everything to go for it and if we don't do it I will take responsibility for it, but I just see it like that."
Chelsea are also in FA Cup action this weekend when they face Manchester City on Saturday.
At the start of this month, Mourinho's side produced a superb tactical display to win 1-0 at the Etihad in the league and the Portuguese says his side will be playing for another win at the weekend.
"There are not two games the same," he added. "Is the approach the same, in the sense we are going to try to win in spite of playing the best team in this country? Yes.
"We don't want a replay, we don't need a replay.
"We are not going there to try to get a draw and to bring the game to Stamford Bridge.
"We want to play, we want to try to win. If we do, we do. If we lose, we lose."