The BBC and The Guardian both claimed that the Brazil-born player, who is currently preparing for the World Cup with Spain in the United States, is now close to completing a move to Stamford Bridge.
Costa, 25, is expected to cost Chelsea around £32 million ($53.6 million, 39.4 million euros).
The rugged centre-forward scored 27 league goals and eight Champions League goals last season for Atletico, who won La Liga and were beaten 4-1 by Real Madrid in the Champions League final in Lisbon.
He notably scored from the penalty spot during Atletico’s 3-1 Champions League semi-final victory over Chelsea at Stamford Bridge.
Costa began his European career with Portuguese club Sporting Braga and initially joined Atletico in 2007.
After loan spells with Celta Vigo and Albacete and a season at Real Valladolid, he returned to Atletico in 2010, spending another period on loan at Rayo Vallecano in early 2012.
His time in Spain enabled him to acquire Spanish nationality and despite having made two friendly appearances for Brazil in March 2013, he decided to commit himself to Spain later that year.
Costa made his Spain debut in a friendly against Italy in March this year, playing for the full 90 minutes in a 1-0 victory at Atletico’s Vicente Calderon Stadium.
He is expected to lead the line for Spain at the World Cup in Brazil, although he is battling an ongoing hamstring problem that forced him off in the early stages of last month’s Champions League final.