Costa has reportedly made it clear that he is suffering from a back problem, only for Chelsea medical staff to dispute his claims.
It is alleged that Conte sided with his staff, prompting a falling-out with the player that has led to him being omitted from the travelling squad for Saturday's Premier League meeting with the reigning champions.
Chelsea declined to comment when contacted by Omnisport but the reports are likely to fuel speculation that Costa could leave Stamford Bridge.
Tianjin Quanjian - who recently signed Belgium star Axel Witsel - have been linked with an £80million bid for Costa, although Conte stressed last week that he had no intention of allowing more players to move to the CSL following the sales of Oscar and John Obi Mikel to Shanghai SIPG and Tianjin TEDA respectively.
Speaking on Friday, Conte hinted at having to make one key decision to his plans before travelling to the King Power Stadium.
"I have to check a couple of situations and then decide," he said. "I don't want to say exactly what and give an advantage to our opponent."
Chelsea, who hold a five-point lead at the top of the table after 20 matches, are looking to return to winning ways following their 2-0 defeat to Tottenham on January 4.