Russian President Vladimir Putin denies having deployed any troops in Crimea and says the soldiers are self-defence groups
US President Barack Obama has urged Russia to move its troops in Ukraine's Crimea region back to their bases and set up a monitoring mission there, White House officials say.
The offer to Russian President Vladimir Putin was made as Moscow remained in de facto control of the strategic region.
Washington has described this as an "act of aggression" by Moscow - a claim denied by Putin.
Meanwhile, Russia said it had tested an intercontinental ballistic missile.
The defence ministry said a Topol RS-12M missile had been successfully launched from Russia's Kapustin Yar test range near the Caspian Sea to the Sary Shagan range in Kazakhstan.
Washington said it had been given advance notice of the missile launch, as required by bilateral arms treaties.
Tensions have been escalating over Russia's military deployment in Crimea and warnings that it could also move into eastern Ukraine to protect Russians and Russian speakers there.
The move has triggered wide condemnation across the globe.
President Obama's offer to Moscow envisaged the return of the Russian troops in Crimea back to the bases of Russia's Black Sea Fleet in the peninsula, the White House officials said.
The plan - which Obama discussed with President Putin on Saturday - also calls for sending a group of international monitors to Ukraine to ensure the rights of ethnic Russians are protected.
The proposal was discussed in details in a phone conversation between President Obama and German Chancellor Angela Merkel on Tuesday.
The Kremlin has so far not publicly commented on the offer.
This comes as US Secretary of State John Kerry and his Russian counterpart Sergei Lavrov are due to meet in Paris later on Wednesday.
Both sides say they would prefer to start a dialogue to ease escalating tensions, and the meeting in France gives them a chance to try, says the BBC's diplomatic correspondent Bridget Kendall.
Moscow has strongly condemned the recent change of government in Ukraine, which came after months of street protests, more than 90 deaths and the flight of ousted President Viktor Yanukovych, a Russian ally.
Speaking during a visit to the Ukrainian capital Kiev on Tuesday, Kerry said there was no indication at all that Russian citizens or Russian-speakers were in any danger in post-uprising Ukraine.
"It is clear that Russia has been working hard to create a pretext for being able to invade further," he said.
In one hint of progress, Ukraine's new Prime Minister Arseniy Yatsenyuk said consultations had taken place between Russian and Ukrainian ministers. He described them as "quite sluggish" but "first steps".
Moscow has tightened its grip over the Crimean peninsula after troops thought to be Russian or pro-Russian began taking control of strategic points on Saturday.
Troops are surrounding Ukrainian military bases and other installations, while two Ukrainian warships are reported to be blocked by a Russian ship in the port of Sevastopol.
Ukrainian TV reported on Tuesday evening that armed men had attempted to take over an anti-aircraft missile base in Yevpatoria, on the coast north of Sevastopol.
Kiev and the West have alleged Russia is mounting an invasion of Crimea, which has a majority Russian-speaking population.
Obama accused Russia of "seeking through force to exert influence on a neighbouring country".
"That is not how international law is supposed to operate," he said.
But at a lengthy news conference on Tuesday, in his first public comments on the issue, Putin denied the heavily armed troops were Russian. He said they were "local self-defence forces" loyal to Moscow, protecting the bases from "nationalists" and "anti-Semites".
Russia, said Putin, reserved the right to act to protect Russian citizens and speakers anywhere in Ukraine.
In Crimea and in eastern Ukraine, there have been shows of support for Russian intervention.
But on Tuesday a peace rally in the eastern city of Donetsk urged Russia to stay away.
"We did not ask for help. I don't want him, Putin, to bring tanks here. I don't want them to shoot at my kids," one woman, Natalia Sytnik, told Reuters.
"Let him hear us: 'Do not defend me from anyone. No-one is attacking me'."
Both the US and the EU have offered financial help to Ukraine, which is facing a growing economic crisis amid its severed ties with Moscow.
Kerry took to Kiev a $1bn (£600m; 720m euros) package of energy subsidies, and told the crowds who remain in Independence Square after months of protests that President Obama "is planning more assistance".
The EU is considering paying the $2bn which Ukraine owes to Russia in gas bills, said EU Energy Commissioner Guenther Oettinger.