It is also claimed the Spain worldwide had a bust-up over an injury with a club fitness coach, in which manager Antonio Conte got involved.
It hasn't stopped the 28-year-old striker from wanting to return to his old club though, as Costa openly admitted he misses Atletico Madrid and wishes he could go back.
Chelsea fans certainly aren't confident that Costa's backup, summer signing Michy Batshuayi, can be as productive.
Birchenall, who played as an attacking midfielder spent time with both clubs including 172 and 75 appearances for Leicester and Chelsea, respectively.
The news has emerged after two Chelsea players already moved to the Chinese Super League during this winter transfer window, with Oscar moving to Shanghai SIPG and John Mikel Obi joining Tianjin TEDA.
Sky Sports is reporting the Brazilian-born Spain worldwide has become unsettled and has not trained for the last three days after he was offered a huge sum to move to the Chinese Super League.
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Conte led Chelsea to victories in all six of their matches last month, the last of which completed a run of 13 wins to equal Arsenal's Premier League record for consecutive wins in one season.
The Daily Mail claim that Chelsea were ready to offer Costa an extension to his current deal, but that is now in major doubt.
Conte is likely to field Eden Hazard as a false nine flanked by Pedro Rodriguez and Willian against Leicester City in Costa's absence, just as he did when the Spain worldwide missed Chelsea's 3-0 win over Bournemouth on Boxing Day due to suspension.
He was also among the players blamed by a section of Chelsea supporters when Jose Mourinho was sacked by the Blues for a second time in December 2015. He was also a disappointment in Sevilla last season with just four goals in 23 games.
Costa has recovered his best form this season and his goals have been instrumental in the club's surge to the top of the table. Unless Chelsea can bring in a truly world-class striker, losing Costa will be a big blow to their title chances.