"There is no controversy, no surprise and no connection with the situation in Georgia or in the region," he said.
The military alliance said the exercises, which will take place just outside Tbilisi, Georgia's capital, will involve about 1,300 people from 19 countries, and are aimed at "improving interoperability between Nato and partner countries".
The countries taking part include Albania, Armenia, Croatia, Georgia, Macedonia and the United Arab Emirates.
A spokesman from Russia's Nato mission said the planned exercises "can only complicate a situation in the region which is not simple to start with".
Relations between the Kremlin and Nato deteriorated after Russia's war with Georgia in August last year, with the military alliance describing Moscow's actions at the time as "disproportionate".
Russia is also opposed to plans by Nato to help Georgia and Ukraine gain membership to the military alliance.
Last month, Nato agreed to resume high-level contacts with Russia, six months after the military alliance had frozen contacts with Moscow over the conflict.
Lavrov also warned Nato members on Thursday not to co-operate with the government of Mikheil Saakashvili, Georgia's president, who is regarded with great suspicion by Russia.
He said: "I hope that the Nato countries will shun steps planned for future cooperation with Georgia ... that could again push the Georgian regime towards an attitude of permissiveness and impunity."