The Spain international - currently representing his country at the FIFA World Cup - scored 36 goals last season as Atletico won a thrilling La Liga title race and reached the UEFA Champions League final.
Diego Simeone's side dumped Chelsea out at the semi-final stage on their way to the showpiece in Lisbon, Costa scoring from the penalty spot in a 3-1 triumph at Stamford Bridge.
The striker dropped a strong hint before the World Cup that he would be heading to Stamford Bridge, saying: "Am I going to Chelsea? It looks like it."
And Mourinho has now revealed that he has demanded Costa be snapped up as Chelsea look to improve upon a trophyless first season with the Portuguese back in charge last time out.
He told Marca: "Costa is a player I want in my team.
"I've asked the club to bring him in and I am extremely confident that we are going to get him and he'll end up signing for Chelsea."
Chelsea have already secured the signature of Cesc Fabregas from Barcelona as Mourinho looks to reinvest cash earned from recent big-money sales of Juan Mata to Manchester United and David Luiz to Paris Saint-Germain.
One player definitely headed to Stamford Bridge from Atletico is goalkeeper Thibaut Courtois - who has spent the last three seasons on loan at the Vicente Calderon.
Courtois is also at the World Cup, with Belgium, and has previously suggested he would prefer not to play understudy to regular Chelsea keeper Petr Cech.
But Mourinho explained: "When the World Cup in Brazil is over, Courtois will be coming straight back to Stamford Bridge - end of story".
