While the authorities say the dogs can be wild and dangerous, reports of their systematic slaughter by a pest removal company hired by the government in recent months have outraged animal rights advocates and cast a gruesome specter over the traditionally cheery atmosphere of the Games.

The handling of the matter has also sharply undercut the image of a friendlier, welcoming Russia that President Vladimir V. Putin has sought to cultivate in recent months.

“We were told, ‘Either you take all the dogs from the Olympic Village or we will shoot them,’ ” said Olga Melnikova, who is coordinating the rescue effort on behalf of a charity called Volnoe Deloe (roughly, Good Will), which is financed by Oleg V. Deripaska, one of Russia’s billionaire oligarchs.