The Telegraph report claims that Costa has not trained with the squad for the past three days after a row with fitness coach Julio Tous, which manager Antonio Conte is also said to have been involved in.
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Costa reportedly clashed with medical staff over the severity of a back injury, according to The Mail:
Costa believes he is suffering from a lower back problem, which could be linked to his history of hamstring injuries. But members of the Blues medical team are not convinced by Costa’s complaints and Conte has subsequently backed his staff.
The 28-year-old will not travel to Leicester with the squad for Saturday's late game, with the Mail adding that his future at the club was now in doubt.
The forward has been unsettled by huge offers from China but it its understood that owner Roman Abramovich will hold the former Atletico Madrid forward to his contract that runs until 2019.
Chinese club Tianjin Quanjian are ready to offer Costa £30 million-a-year, a report from Sky Sports claims, with the Chinese Super League side ready to tempt Chelsea a transfer fee of £80 million.
Tianjin Quanjian are managed by World Cup winner Fabio Cannavaro, and recently signed Axel Witsel from Zenit St Petersburg.
It has since been reported by Simon Johnson of the London Evening Standard that Costa's agent Jorge Mendes met with the chairman of TIanjin on Friday.
Chelsea have so far declined to comment.

OUR VIEW

If you were going to bet on any player being involved in something like this you'd bet on it being Diego Costa. However this comes after Costa has publicly stated how happy he is in England, and working under Conte.
He's the Premier League top scorer and he looks in fantastic shape this season, he's even stopped picking fights.
However the money that is being quoted is absolutely incredible. He could increase his wages seven-fold and he's only got a few years left in his career before he'll probably have to think about winding down, especially with the intensity of his game.
It seems as if the row has come after his 'head has been turned' by the offer from China according to reports. It's tough to know what Chelsea do from here.
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