The Daily Star says Costa is currently in Brazil without his club's permission and was fined two weeks' wages - around £280,000 - on Monday.
If he stays AWOL, he stands to lose around £3m in wages.
It is understood they can argue his absence, while he is still under a two-year contract with them, devalues him as an asset.
In the current market, Costa, 28, would be worth at least £50m. That value could drop by around £10m, insiders estimate.
One expert, who asked not be named, said: “I definitely do think they can take him to court.
“I think there are a lot of lawyers out there who are quite keen to take it on."