On Friday morning Wenger 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 Premier League this season, describing his side as the "little horse" in the race and making them third favourites behind Manchester City and the Gunners, despite currently leading the standings.
"He is a specialist in failure. I'm not," said Mourinho, whose side play City in the FA Cup fifth round on Saturday.
"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."
As defending champions, the title is Manchester United's to lose, not leaders Chelsea's, Mourinho argues.
Mourinho is Chelsea's most successful manager after twice winning the Premier League in his first spell and was reappointed by Blues owner Roman Abramovich last summer on a four-year contract, which the Portuguese hopes to extend, but only on merit.
"I don't want eight years," Mourinho said.
"I just want my contract of four years to do my work and after that (I will) deserve or not deserve a next contract."
After delivering his riposte to Wenger, Mourinho was reluctant to discuss further the Premier League title race, with his attention on the FA Cup.
Mourinho's most recent trophy as Chelsea boss was the 2007 FA Cup and his side could end Manuel Pellegrini's quest for an unprecedented quadruple with a second successive victory at the Etihad Stadium on Saturday evening.
Chelsea were expected to "park the bus" in a defensive performance at the Etihad Stadium last week, but surprised with a counter-attacking display which earned a 1-0 win courtesy of Branislav Ivanovic's goal.
Mourinho, who is without key defenders John Terry (gluteal muscular strain) and Gary Cahill (calf), will not alter that approach.
"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."
Chelsea have twice beaten City in the league this season and Mourinho has experienced defeat against Pellegrini just once in nine meetings in Spain and England.
"I don't like this histories of records between managers, because it's between teams, it's not between managers," Mourinho said.
The proximity of the recent triumph makes the FA Cup challenge difficult.
"It's not easy to play consecutive matches against the same team, exactly because you try to read the previous game and try to improve things and try to analyse the mistakes and try to analyse the opponent again," Mourinho added.
"The game was very equal. Are we confident? Yes, but I believe they are confident.
"Are they worried? I believe so, but we are too."
Mourinho's concerns increased when Cahill failed to come through training on Friday.
"He tried to train," Mourinho said.
"He didn't finish the training session in the best conditions. He's going for a scan."
Ivanovic is likely to be deployed at centre back, Cesar Azpilicueta at right-back and Ashley Cole will come into the side at left-back.
Cole has found his opportunities limited by the form of the Spain right-back on his 'wrong' flank but enjoys the FA Cup, having won the trophy a record seven times.
With the Premier League's top four doing battle - Chelsea play City and Arsenal face Liverpool - Mourinho believes the tournament represents a good trophy chance for the smaller clubs.
Mourinho said: "Normally people like the big finals, but the reality is that the draw puts things in a way where the final can be unexpected.
"That's the beauty of the Cup. For all of them it's a fantastic opportunity."