The Chelsea coach hit back at Wenger, who had earlier said that the title was Chelsea's to lose and that Mourinho had ruled his side out of the race because of a fear of failing.
Mourinho, who claimed last week that everything he says and does is "mind games", returned fire later.
"He is a specialist in failure," Mourinho told reporters.
"If supposedly he (Arsene Wenger) is right and I am afraid of failure it is because I don't fail many times.
"He is specialist because eight years without silverware is very long. If I do that with Chelsea, I leave and I don't come back."
Chelsea, who face Manchester City in the fifth round of the FA Cup on Saturday, are top of the Premier League, a point clear of Arsenal in second.
Mourinho, who often provokes and antagonises other managers with barbed comments, turned his guns on Wenger after last week aiming volleys at Manchester City and their manager Manuel Pellegrini.
After each manager labelled the other team as being title favourites by virtue of how much they had spent in the transfer market, Mourinho hinted City were not operating within the same UEFA Financial Fair Play guidelines.
He then turned his attentions to City's Ivory Coast midfielder Yaya Toure, who aimed an off-the-ball kick at Norwich striker Ricky van Wolfswinkel in their 0-0 stalemate last weekend but escaped any FA punishment.
Chelsea face City just 12 days since the London club won 1-0 at the Etihad Stadium in the league.
"It is not easy to play consecutive matches against the same team. You try to read the previous game and try to improve. Sometimes you can be influenced by the direction the game takes," Mourinho added.
"We know they (Manchester City) are strong. Our priority is always the next match."