The two players were signed with great fanfare last month by Persib Bandung, one of the country’s top clubs, as Indonesian football looks to launch a new league and emerge from a FIFA suspension.
But the government-backed Professional Sports Agency found Essien and Cole did not have the necessary work permits and they could now face questioning by immigration authorities.
The club admitted on Monday it did not have the necessary paperwork, despite both players making their debuts in Liga 1 at the weekend.
Persib said it began processing the paperwork days before the match, but the permits were not ready in time and the football association and league organiser gave them permission to play.
Manager Umuh Muchtar accused the sports agency, which monitors foreign players in the country and reports to the government, of going too far.
He said: “That they are here is a joy for us, they want to raise up Indonesia’s name in the world.
“Everybody knows that Essien is a world-class player, not an illegal immigrant.”
Immigration authorities summoned the club management on Monday.
A spokesman said: “They (the players) are professionals so they must hold a permit to perform activities in Indonesia.”
Essien, 34, is Ghanaian and has played for Chelsea, AC Milan and Real Madrid.
Cole, 33, played seven times for England and was a striker for Chelsea and West Ham.