Chelsea have long been linked with a big-money move for the 25-year-old, who netted 36 goals in all competitions last season to help Atletico win the Primera Division title and reach the Champions League final.
Mourinho told Spanish newspaper Marca: "Costa is a player who I want to have in my team. I've asked my club for him and I have total confidence that we're going to achieve it and he will end up signing for Chelsea."
Chelsea have already started their summer rebuilding after failing to win a trophy last season, with the club this week announcing the signing of Cesc Fabregas from Barcelona as well as completing the sale of David Luiz to Paris St Germain.
Several other first-team players are also set to leave the west London outfit this summer, but Mourinho insists midfielder Ramires will not be one of them.
Reports in Spain suggest Real Madrid's former Chelsea boss Carlo Ancelotti is keen on taking the Brazil international - who the Italian signed from Benfica in 2010 when he was in charge at Stamford Bridge - to the Bernabeu this summer.
However, Mourinho is adamant the 27-year-old is not going anywhere, saying: "Ramires is staying at Chelsea, he's not for sale.
"I have a great relationship with important people at Real Madrid and they know that Ramires is going to stay at Chelsea."
The Portuguese, who left Madrid last summer to return to Chelsea, also rubbished speculation he was looking to raid the European champions for any of their players.
Mourinho added: "They (Real Madrid) know that Ramires isn't going, but they also know that I'm not interested in any Real Madrid player. (Raphael) Varane and (Sami) Khedira? I repeat that I'm not interested in any Madrid player, full stop."
Mourinho also confirmed to the paper that Belgium goalkeeper Thibaut Courtois would be returning to Chelsea after the World Cup having spent the last three seasons on loan at Atletico.
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