Press Association Sport understands Costa did not travel with the rest of the squad for the Saturday evening fixture.
The striker, however, expressed his support for his team-mates late on Friday night by posting a photo on Instagram from the Blues' win over Tottenham in November, along with the caption: " Come on Chelsea!!!", plus a heart emoji
Chelsea declined to comment on reports that the 28-year-old had had a disagreement with head coach Antonio Conte and his coaching staff and has been training alone this week.
Conte on Friday afternoon said he had some undisclosed doubts for the game, but mentioned no names or reasons for potential absences.
He was also asked about Dimitri Payet telling West Ham he wanted a move.
"I read about this situation," Conte said.
"It's very difficult for me to talk about another player, of another team.
"For my club, for my players to solve our problem if there are problems, it's important."
Costa has been prolific this season for a Chelsea side five points clear at the top of the table entering this weekend's fixtures, scoring 14 goals in 19 Premier League games.
Earlier this month he was linked with a move to the Chinese Super League, with the accompanying astronomical wages.
It was rumoured Tianjin Quanjian were prepared to bid Â£80million and Costa's future could be the main talking point of the winter transfer window, which closes on January 31.
Costa signed a five-year deal on his arrival from Atletico Madrid in July 2014 for Â£32m, so has two and a half years to run.
Chelsea owner Roman Abramovich would resist Costa agitating for a move.
Replacing such a prolific goalscorer in January would be challenging and Costa's understudy Michy Batshuayi is yet to make a Premier League start.
Costa was repeatedly linked with a return to Atletico after being troubled by hamstring problems and disciplinary issues in his first two seasons at Stamford Bridge
He recently admitted he had wanted to leave, before stating he was happy.
Costa was many pundits' player of the championship at the halfway mark.
His understanding with Eden Hazard and Cesc Fabregas, in particular, had troubled opponents to the extent Chelsea won 13 straight games until the loss at Tottenham on January 4.
Hazard played centrally, with Pedro and Willian either side, when Costa was suspended for the Boxing Day win over Bournemouth
It is likely Conte will turn to Hazard once again.
The trip to Leicester is Chelsea's first Premier League game since the defeat at Spurs and comes after an FA Cup third-round win over Peterborough, when Costa was an unused substitute after Conte made nine changes to rotate his options.
Former Liverpool and Newcastle manager Graeme Souness reckons Chelsea will not win the title if they lose Costa.
"They cannot afford to lose him," Souness told Sky Sports News.
"From the club's side it's the very last thing Chelsea need
He's the leading scorer in the Premier League now - lose him and they aren't going to win the Premier League
That's how important he is."
Source : PA