Mr Rogozin says holding the exercises in Georgia would be "impossible"
Russia has threatened to pull out of a meeting with senior Nato commanders if the alliance goes ahead with planned exercises in Georgia next month.
Russia's ambassador to Nato, Dmitry Rogozin, said its president and foreign minister had raised concerns and that the government was awaiting a response.
"If there is no reaction, we shall take certain steps," he told Russian TV.
Nato says the exercises were planned before last year's conflict between Russia and Georgia over South Ossetia.
The war led to Nato temporarily cutting high-level contacts with Moscow.
Georgia hopes eventually to join Nato, a move strongly opposed by Russia, which says the alliance's eastward expansion is a threat to its security.
Last week, Russian President Dmitry Medvedev described as dangerous and wrong Nato's decision not to postpone the military exercises in Georgia from 6 May to 1 June, which are expected to involve 1,300 troops from 19 countries.
"Such decisions are disappointing and do not facilitate the resumption of full-scale contacts between the Russian Federation and Nato," he said.
His foreign minister, Sergei Lavrov, had earlier warned that the exercises would not help develop stability in the Caucasus.
On Monday, Russia's envoy to Nato said that if it did not receive a response from the alliance addressing its concerns, the meeting of the chiefs of general staff of the Russia-Nato Council scheduled for 7 May would not take place.
"We are waiting for reaction - and even more so, we are waiting for reaction after both the foreign minister and, especially, the president of the Russian Federation have made statements on this subject," Mr Rogozin told Vesti TV.
"Therefore, if there is no reaction, we shall take certain steps."
Nato has said the exercises will be non-aggressive, involve no heavy armour, and be based on a fictitious UN-mandated, Nato-led crisis response operation.
But Mr Rogozin said last week that holding the exercises in Georgia would be "impossible" and that they exercises could be exploited by President Mikhail Saakashvili in his stand-off with opposition parties.
The opposition accuses him of mishandling the war with Russia, during which Georgia's attempts to regain control of the breakaway regions of South Ossetia and Abkhazia were repelled by Russian forces.
Under an EU-sponsored ceasefire, monitors were sent to Georgia. But thousands of Russian troops remain in both breakaway regions.