Brendan Rodgers' side made a £21.8 million bid for the Brazilian forward last week, but the Madrid club insisted he was not for sale.
Now Atletico's sporting director Jose Luis Caminero has reiterated the club's stance on the 24-year-old, who scored 20 goals in all competitions last season.
Caminero was quoted by Spanish daily AS as saying: "Diego Costa is a very important player for Atletico, we value him very highly and we are convinced that this year he is going to score a lot of goals with Atletico. He is an idol for the fans."
Pressed on whether Costa would stay at the club, he added: "Yes, he's going to continue."
Caminero's comments follow Costa's own declarations that he was intent on remaining at Atletico, who finished last season in third place in La Liga. The club qualified automatically for the Champions League, and won the Copa del Rey, with Costa scoring in the final against rivals Real Madrid.
Speaking at Madrid Barajas airport as the team returned from their pre-season tour of South America, Costa said: "As I've been in Argentina and Uruguay I've not heard anything about this.
"I'm happy here, if the club want me to be here, I'm going to be here.
"I think that the coach (Diego Simeone) counts on all of us, not just me. He knows how important it is to maintain the squad, the fans know it as well."
The club are reluctant to lose Costa after Radamel Falcao, their top scorer last term with 28 league goals, joined Monaco at the start of the summer.
Spain's all-time top scorer David Villa was signed as the Colombian's replacement, while 20-year-old forward Leo Baptistao has joined from Rayo Vallecano.
Baptistao scored a double in Atleti's win over Uruguayan club Nacional in their last game of their tour, but Villa did not travel with the squad due to his participation in the Confederations Cup with Spain.
Caminero explained that 'El Guaje' would need time to return to full fitness and adapt to his new club, but was certain he would be a success.
"He finished the season later than everyone else and he needs a different plan of preparation. As soon as we return to training he will join up and be part of the squad," he said.
"He will have ideal forward partners like Adrian or Diego Costa. We don't need to remind ourselves how good a player Villa is, at international and at club level. I'm sure that his goals will take Atletico as far as possible and we can assure everyone that he has the utmost enthusiasm about the coming season."