The two were signed with great fanfare last month by Persib Bandung, one of the country's best clubs, in a boost for Indonesian football as it seeks to bounce back from a FIFA suspension and launch a new league.
But the government-backed Professional Sports Agency found the players did not have the required work permits.
The club admitted they had not finished processing the paperwork for the pair before they made their debuts in the country's new top-flight Liga 1 at the weekend.
After calling in the players and Persib's management for questioning, immigration authorities said Tuesday the pair would not be able to play until they obtained the permits.
"Today we gave a strong warning to Persib in the form of a letter saying that as long as Persib had not completed the permits, which should be issued by the manpower ministry, we are banning Essien and Cole from playing," said Maulia Purnamawati, the immigration chief in Bandung, the city where the club is based.