"What a journey these last two years! Every day we became stronger, more determined, with more will and sacrifice, managing to break the mediocrity that we are often surrounded by," Conte wrote on Instagram.
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.
"This is great news, I wish him all the best - apart from when he plays against Inter," Conte told Italia 1.
Inter were hit with a COVID-19 outbreak three weeks ago that meant their final league match before the international break, against Sassuolo, was postponed.
Stefan de Vrij and Matias Vecino tested positive for the novel coronavirus, joining captain Samir Handanovic and Danilo D'Ambrosio in quarantine at home.
Champions League rivals Atalanta and Napoli go head-to-head on Sunday before the Bergamo side host Real Madrid in this year's elite European tournament.
Inter, missing top-scorer Romelu Lukaku through suspension, should have equalised 11 minutes after half-time, but Alexis Sanchez's shot was brilliantly cleared off the line by Merih Demiral.