A Chelsea statement on Thursday said that they had agreed terms with the La Liga side for the transfer of Costa, with Spanish newspaper El Mundo saying the fee would be 55 million euros (£48.66 million).
When asked whether his relationship deteriorated with the Spanish striker over the last few months, Conte chose to focus on the future of the club starting with a league trip to Stoke City on Saturday.
"We want to thank him (Costa) for what he did with this club. We wish him all the best for the future," the Italian told reporters.
"The past is not important. If you did well in the past it's okay but we have to work for the present and above all for the future.
"I don't forget that we won together last season. I repeat for sure we want to thank him for his effort last season, not only last season but the period he played for Chelsea."
Conte believes Chelsea face one of their toughest challenges of the campaign at the bet365 stadium, after Mark Hughes's side picked up four points from two games against Arsenal and Manchester United.
"For sure it will be a really tough game. As you said before they beat Arsenal and drew against United, it means they are a very good team," Conte said.
"We must pay great attention tomorrow. This is the first of three tough games in seven games. A good test and challenge for us."
Chelsea have a busy schedule next week, starting with Wednesday's trip to Atletico in the Champions League before hosting league leaders Manchester City three days later.
"This season it will be a big challenge for us, to face each competition with our squad and play every three days," he added.
"This type of situation will be very important for us to face in the right way. It won't be easy to have these games very close."
(Reporting by Hardik Vyas in Bengaluru; Editing by Christian Radnedge)
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