The United boss rose to the bait after Conte told his Chelsea players to avoid a “Mourinho season” following their title triumph last year.
Mourinho was sacked by Chelsea just months after lifting the Premier League trophy in 2015 with the club struggling in the bottom half of the table.
How Manchester United could line up next season
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"Two years ago Chelsea finished 10th. That cannot happen again," Conte said.
"We know the difficulty and for sure we want to avoid the Mourinho season with Chelsea.
"The coaches of the previous two champions [Chelsea and Leicester] both lost their jobs [the next season] and we are working very well to find the best solutions and use them.”
"I could answer in many different ways but I am not going to lose my hair to speak about Antonio's comments,” Mourinho said.
Conte struggled with balding as his playing days ended but arrived at Stamford Bridge with a full head of hair after reports that he had received three rounds of transplant surgery.
A picture of the Serbian midfielder in United training kit appeared over the weekend, feeding rumours that the deal was completed, but Mourinho admitted that was not quite the truth.
"I am waiting for news," Mourinho said in a press conference.
"I know that he wants that very, very much. When a player wants very, very much, the chance is bigger.
"I think we have a chance but in football until it's official, I saw so many things happen that I refuse to say more than I am telling you now.”
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