Ukraine began a tense countdown Monday to weekend elections as the country’s richest man called for a huge peace rally in the restive east and Russian President Vladimir Putin said troops deployed near the border have been ordered home.

Despite Putin’s announcement, NATO Secretary General Anders Fogh Rasmussen said the alliance has “not seen any evidence at all that the Russians have started withdrawal of troops from the Ukrainian border.”

Putin previously claimed that about 40,000 Russian troops massed along Ukraine’s border had been withdrawn, but the United States and NATO say they have seen no signs of that. Putin’s office issued a statement early Monday saying troops involved in “routine spring” exercises in the Rostov, Belgorod and Bryansk regions had been ordered back to barracks.

In an interview Monday, Pavlo Sheremeta, Ukraine’s economy minister, welcomed Putin’s announcement, saying it appeared to go further than the Russian leader’s earlier declarations about withdrawing troops. Sheremeta said he sensed what could be a shift in tone and, hopefully, in substance. “Is this credible? We shall see, but at even up until 10 days ago, all his declarations were much more hawkish,” he said.

Meanwhile, Rinat Akhmetov, whose coal mines, steel plants and other factories are the industrial might of the Donets Basin in eastern Ukraine, said in a statement late Monday that he had asked all employees to stage a peaceful “warning protest” at their companies to send the message that a separatist rebellion endangers the region’s economy.

Akhmetov said he called for the demonstration at noon Tuesday after intervening to halt employees’ plans to hold a peace rally in Mariupol on Monday. Akhmetov said he suspended Monday’s rally after he learned that unidentified gunmen had planned to shoot anyone who took part.

Ukraine is set to go to the polls Sunday in historic presidential and mayoral elections that could affect the country’s very makeup and its alignment between the West and Russia. Tensions remained high in the troubled Donetsk and Luhansk regions near the country’s border with Russia.

Pro-Russian militants, who have seized administrative buildings in eastern Ukraine and fought government troops in the industrial region, have said they will boycott the elections.They allege that the recent chaotic referendums on self-rule in eastern Ukraine were valid. Kiev called the referendums a farce, and the West declared them illegal. Russia said their results — in favor of self-rule — should be respected.

Kunkle reported from Donetsk. Abigail Hauslohner in Moscow and Griff Witte in London contributed to this report.