Moscow believes that the planned exercises in May "can only complicate a situation in the region which is not simple to start with", The Daily Telegraph reported.
A spokesman said the plans were "absurd" in the light of tensions with the breakaway Georgian regions of Abkhazia and South Ossetia – which Russia has recognised as independent – as well as opposition protests against Georgian President Mikheil Saakashvili.
"We will send a letter to Nato calling for them (the exercises) to be postponed, at least for the time being," he added.
Earlier Nato announced that it would hold exercises in Georgia from May 6 to June 1 to improve co-operation between the alliance and partner countries.
The exercises, which have been planned since the spring of 2008, will involve about 1,300 people from 19 Nato and partner countries and will be held at a training centre 20 kilometres (12 miles) east of Tbilisi.
The military alliance said in a statement that the exercises had the "aim of improving interoperability between Nato and partner countries."
Nato spokesman Robert Pszczel said that so far Nato Secretary General Jaap de Hoop Scheffer had received no request for a postponement from Russia.
He added that they would be essentially co-ordination exercises, involving "no heavy military equipment."
"There is no controversy, no surprise and no connection with the situation in Georgia or in the region," he stressed.
Georgia is not a member of the alliance. In Bucharest last April, Nato leaders rejected granting Georgia and Ukraine candidate status, but agreed they had a future in the alliance in the face of deep opposition from Moscow.
Relations between Moscow and Nato hit a low point after Russia`s brief war with Georgia last August, after which Moscow formally recognised the two rebel regions.
Nato has condemned that recognition as a violation of Georgia`s territorial integrity.
However ties have improved this year, not least because the new US administration said it wanted to "press the reset button" in its relations with Moscow.
US President Barack Obama has announced that he will visit Moscow in July.
The US and Russia have also agreed to start talks on cutting their nuclear arsenals.