Wenger yesterday morning declared the title is Chelsea's to lose and insisted anyone who claims their team are not in the race does so out of fear of failure.
Mourinho has repeatedly played down Chelsea's chances of winning the Barclays Premier League this season, insisting that his side are third favourites behind Manchester City and Arsenal, despite being leaders with 12 games remaining.
"He is a specialist in failure. I'm not," said Mourinho, whose side play City in the FA Cup fifth round today. "If supposedly he is right and I'm afraid to fail, it's because I haven't failed many times. Maybe he's right. Maybe I'm not used to failing.
"The reality is he is a specialist. Eight years without a piece of silverware, that's failure. If I do that at Chelsea, eight years, I leave London and I don't come back."
During his first spell, Mourinho described Wenger as a "voyeur" for repeatedly commenting on Chelsea.
It is a choice of words now regretted by the Portuguese, but he was riled by Wenger's most recent comments. "He loves to look at this football club," Mourinho added. "I thought between 2007 and 2013 was enough time for him to forget this.
"But it looks like he always likes to look at us. He is saying we're not candidates because we're afraid of failure. But failure of what? If we don't win a title in one year, is that a failure? Or if we don't win a title in two years, is that a failure?
"I have lots of respect for him and I think he is a great manager. But when he speaks about failure . . . failure is not winning a title in seven or eight years. That's failure.
"If I don't win a trophy in four years, I don't want a new contract, even if the club wants to give me [one]. It's as simple as that.
"I don't think a manager should be embarrassed when a manager gives everything. If you don't get results, that's football. But for my mentality, there is a limit. And you have to be strong enough and proud enough to admit when it's enough."
After delivering his riposte to Wenger, Mourinho was reluctant to discuss further the title race, with his attention instead on the FA Cup. Chelsea have twice beaten City in the league this season and Mourinho has experienced defeat against Manuel Pellegrini just once in nine meetings in Spain and England.
Pellegrini recently stated he did not wish to talk about Mourinho, who insisted he would prefer not to talk about his counterparts.
Mourinho added: "If he [Pellegrini]doesn't like to speak about me, perfect. And hope he does what he says. I also don't like a lot to speak about other managers."
Mourinho's most recent trophy as Chelsea manager was in the FA Cup in 2007 and his side could end City's quest for a quadruple with a second successive victory at Etihad Stadium on Saturday evening.
Chelsea were able to subdue the influential Yaya Toure in the win earlier this month. The Ivory Coast midfielder is free to play after avoiding a Football Association suspension for an off-the-ball kick on Norwich's Ricky van Wolfswinkel, but Mourinho is not focused on him.
"They are good at everything," Mourinho said. "Is Yaya Toure important? Yes. He's so important that the FA decided he has to play this game."