The leaders have agreed to continue the discussion on Saturday morning, he told reporters.
Appathurai would not say whether the front-runner, Danish Prime Minister Anders Fogh Rasmussen, was the only candidate. Nor did he say whether Turkey was the only NATO country that has voiced concern over the candidacy of Rasmussen.
It is reported that Turkey is concerned that Rasmussen would not be proper for the job as NATO's current top operation in Afghanistan involves engagement with the Muslim world. The publication in Denmark of cartoons about Islamic prophet Muhammad in 2005 sparked violent protests in several Muslim countries.
The current NATO secretary general's term of office expires on July 31 this year