SOCHI, Russia, Feb 4 (Reuters) - An outcry over the fate of stray animals being rounded up in Sochi is the latest problem to dog Russia's preparations for the Winter Olympics.

Already scrambling to get hotels ready on time, make the Games safe and convince the world Russians are not homophobic, the host nation is now trying to shake off accusations that it is killing stray dogs.

It is hardly the showcase for modern Russia that President Vladimir Putin had hoped for at the Olympics.

Residents say thousands of dogs have disappeared from the streets of Sochi since a local business won a contract to catch stray animals before the Games, which open on Friday.

Although some can still be seen wandering the streets, including close to some of the sports venues, reporters and local residents say there have been fewer each day.

The Games organisers are reluctant to discuss the dogs' fate, particularly whether they have been culled.

"There is a service that catches the dogs and the city authorities handle this. The city has a special shelter for them," said Alexandra Kosterina, a Games spokeswoman.

The city authorities did not immediately comment but the Basya Services pest control company said it had received a contract to catch and destroy the wild dogs roaming the Black Sea resort.

Residents say many of the dogs on the loose had been kept as pets or guard dogs by foreign workers who helped build the Games venues but have now returned home. Some of the dogs, they say, have been found dead in the streets.

Tatyana Leshchenko, an animal activist, told local media: "The city authorities are paying for the murder of each animal but this does not solve the problem. This money should be provided for sterilising animals and building shelters."

(Reporting by Timothy Heritage; editing by Keith Weir)