Tianjin Quanjian, who have signed Belgium midfielder Axel Witsel, are in the market for a striker and have been linked with an £80m bid for Costa.
Chelsea had been hopeful of agreeing a contract extension with Costa after the player had indicated that he was far happier with life in England, but the latest issue raises new doubts.
Costa is famed for his combustible personality and his moods swing so dramatically that he can change from appearing to be desperate to leave to being happy once again in the space of a few days.
Chelsea will hope this latest episode quickly blows over, but it has given Conte a major headache as he attempts to get his team on to another winning run after he won the manager of the month award for December and set a Premier League record by taking the trophy for three successive months.
Should Conte decide to punish Costa by leaving him out at Leicester, then he is likely to play Eden Hazard through the middle with Pedro Rodriguez and Willian wide, as he did when Costa was suspended for the game against Bournemouth that Chelsea won 3-0.
Costa this week posted a series of pictures and videos of him training on his own that could be seen as a reaction to the training ground incident with Chelsea's fitness guru.
Bertelli himself can be an outspoken character and was threatened with being sent to the stands in the defeat at Tottenham for constantly shouting at the fourth official.
Costa is the Premier League's top scorer with 14 goals and had looked to have been tamed by Conte for a long period this season, when he went 11 games without being booked.
The pair have also struck a close personal relationship, despite the fact Costa showed his frustration at being shouted at by Conte in the victory over Leicester at Stamford Bridge by signalling that he wanted to be substituted.
Costa was warned not to repeat that behaviour and it seems Conte is now considering taking action after this week's fall-out.
Conte played down an incident between Costa and Pedro in the defeat at Tottenham, when Costa shouted at his team-mate for not making the correct run after a chance went begging.
There have also been reports of other instances in which Costa has let his passion get the better of him in training, but they have all been quickly dealt with.
Conte hinted that he had at least one big decision to make for the Leicester match by saying: "I have to check a couple of situations and then decide. I don't want to say exactly what and give an advantage to our opponent."
Asked about whether he has ever had a player refuse to play for him in relation to Dimitri Payet going on strike at West Ham United, Conte replied: "It's very difficult to talk and above all for me it's very difficult for me to talk about another player. For my club, for my players, we solve our problem if there are problems. This is important."
Despite previous flashpoints under Conte this season, Costa's behaviour had improved markedly until the latest incident. Last term, the former Atletico Madrid star tossed a bib in the direction of former manager Jose Mourinho and was involved in a training ground fight with Oscar, who has now moved to China.
Costa has admitted that he wanted to re-join Atletico last summer, but was talked into staying by Conte and it now remains to be seen whether he can once again be convinced to ignore outside interest and regain his Stamford Bridge focus.
The headache over Costa could also influence Conte's decisions over strikers during the January transfer market. He is interested in signing Fernando Llorente from Swansea City and Chelsea still hope to keep Michy Batshuayi at the club.
Batshuayi is a loan target of West Ham, his former club Marseille, Paris Saint-Germain and Lille, but Conte will be even more reluctant to let him go if there are any lingering problems surrounding Costa.
Conte has insisted that captain John Terry will stay at Chelsea this month, despite interest from Bournemouth, saying: "I speak with him every day and John never talked to me about this situation. It means that he is very happy to stay here and, above all, I am very happy that John stays here and helps me."
