Konsa, 27, has played with the national side 20 times but was not in the squad for two international friendlies against Oman and the United Arab Emirates this month.
The striker was arrested by police on Monday with three others suspected of distributing cocaine. Konsa has not yet been charged.
Nomme Kalju, Konsa's club who are 2012 Estonian champions, suspended the player immediately.
"It is very sad. We are shocked by this news," said Nomme Kalju sporting director Sergei Terehhov.