Conte led Inter to their first Serie A title in 11 years on Sunday with four games to spare, before Mourinho was surprisingly announced as Roma's new coach from next season two days later.
There is a long-standing history of animosity between the pair, who exchanged barbs in the Premier League while Conte was in charge of Chelsea and Mourinho was at Manchester United from 2016 to 2018.
"This is great news, I wish him all the best – apart from when he plays against Inter," Conte told Italia 1.
Moyes: Clubs have missed out on Rice 'bargain'
"There is respect between us. But when you are in competition, the Latin saying applies: 'mors tua vita mea' (your death, my life)."
"Make no mistake, Jose Mourinho is walking into an extremely tough job," writes Pete Sharland in "Out of the Mou - Will Jose's latest move work?".
"Fonseca did a great job rebuilding Roma after the failed Monchi experiment but they are lagging behind the rest of Serie A. With four games to go they are 27 points behind new champions Inter Milan. They are 14 points off Milan in the Champions League spots and perhaps worst of all, nine points behind arch-rivals Lazio.
"The club of Francesco Totti and Daniele De Rossi is not just the romantic runner-up, they are no longer in the conversation."
Inside Europe: Who will win El Clasico? Has game lost lustre? Which youngsters could star?
Euro round-up: Muniain claims win for Athletic against Villarreal, Leipzig trail Bayern and Dortmund