Russian President Vladimir Putin has ordered troops he said were staging military drills in areas near Ukraine to return to their home bases.
The Kremlin made the announcement Monday, just days before a crucial presidential election in Ukraine.
Western countries have protested the deployment of tens of thousands of Russian troops near the Ukraine border, expressing concern the deployment might be preparation for a land grab after Russia annexed Crimea in March.
NATO said on Monday it has yet to see any signs of troop movements.
"Now I think it's the third Putin statement on withdrawal of Russian troops, but so far we haven't seen any withdrawal, at all," said NATO Secretary General Anders Fogh Rasmussen.
He added that an actual withdrawal of Russian troops would be a first important contribution to de-escalating the crisis.
The Pentagon, too, has said it has seen no signs of a withdrawal.
Moscow calls on Kyiv
Moscow called on Kyiv on Monday to immediately withdraw its troops from eastern Ukraine, where they have been battling a pro-Russian insurgency.
“Russia calls for an immediate cessation of the punitive operation, all violent actions, a withdrawal of troops and for the resolution of all existing problems only by peaceful means,” a Kremlin statement said.
Ukrainian authorities say they are fighting “terrorists” and “separatists” in the eastern part of the country.
Armed pro-Russian rebels in two of Ukraine’s eastern regions have declared their independence from Ukraine and voiced support for joining Russia. Violent skirmishes have broken out in numerous eastern cities between Ukrainian security forces and the insurgents.
US, EU warn Russia on election interference
The United States and its European allies have warned Moscow that they will impose new sanctions against key economic sectors of the Russian economy if it disrupts an presidential election to be held on Sunday.
The Ukraine Central Election Commission on Saturday said the unrest could prevent almost two million people from voting.
Ukraine’s prime minister, Arseniy Yatsenyuk insisted on Monday that balloting will be held throughout the country.
“We are aware that in some places polling will be difficult but there won’t be many such places, and this will not impact the election results,” Interfax-Ukraine quoted him as saying.
Yatsenyuk said the vote will be legitimate and Ukraine will get “a legitimately elected president.”
Ukrainian confectionery tycoon Petro Poroshenko commands a decisive lead in pre-election opinion polls, and could secure the more than 50 percent of votes needed to win in the first round.
Lavrov strikes conciliatory tone
Speaking in Moscow Monday, Russian Foreign Minister Sergei Lavrov struck a somewhat more conciliatory tone on the Ukraine crisis and tensions it has produced between Russia and the West.
He said Moscow's ties with the European Union and NATO needed a "rethink'' in light of deep differences over the crisis:
"These relations require a substantial rethink, and together with our partners from the EU and NATO nations we are trying to conduct an analysis in order to better understand where we are, where our assessments coincide and where we disagree, and what to do to bring those relations back, those documents that form the basis for Russia and NATO cooperation, as well as European Union and Russian engagement,'' said Lavrov.
Lavrov also said the so-called round table discussions that began last week with Ukrainian ministers, regional deputies and business leaders, while imperfect, were a good start toward resolving the crisis in Ukraine.
He called on the Organization for Security and Cooperation in Europe to mediate in the effort.
Separately, President Putin spoke with German Chancellor Angela Merkel by phone on Monday, with both agreeing that the crisis in Ukraine should be resolved peacefully, a Kremlin statement said.
Odessa violence probe
Foreign Minister Sergei Lavrov has called on the OSCE and the United Nations to ensure an “open and impartial” probe by Kyiv into deaths that occurred in the Ukrainian city of Odessa earlier this month, the Foreign Ministry said on Monday.
At least 37 people in the southern port city died after a riot on May 2 ended with dozens of pro-Russia protesters being trapped in a burning building.
Lavrov’s call follows an announcement by Ukrainian officials that the protesters might have been poisoned with chloroform before the fire broke out.
The chief of Ukraine's General Investigative Directorate, Vitaliy Sakal, told reporters earlier on Monday that chloroform had been found by investigators in the charred remains of the Trade Unions building in Odessa. Inhaling the substance causes breathing failure.
Sakal added that Ukrainian investigators have contacted the Israeli Embassy, asking for professional experts to help investigate any traces of the chemical in the building.
The circumstances under which the fire broke out and who is to blame for the protesters’ deaths remain in dispute.
VOA's Jeff Custer contributed to this story. Some information for this report was provided by AFP, AP, RFE/RL and Reuters.