Footage of the incident showed a fresh landslide on an embankment, which made a several-metre-deep hole near the stadium and covered a sidewalk with mud.
A spokesman for the Sport In company, overseeing the construction, said there were also leaks inside the stadium which could take up to a week to repair.
“I’m sorry for the city. They invested so much money and it was washed away by one rain,” 19-year-old Vitaly Ovchinnikov, a local resident, told Reuters TV.
Volgograd Arena, a stadium that can accommodate up to 45,000 spectators, was built for the World Cup at a cost over 16 billion roubles (194.20 million pounds).
“It’s not normal. It was built with federal money, with taxes we pay them,” another local resident, Mikhail Nesterenko, 64, said.
Eight teams played World Cup group matches in Volgograd, including England and Japan, which made it to the knockout stages.
Spokesmen for the Volgograd government and Sport In said the rain was extraordinary and was dealt with as an emergency.
“This happens once in 100 years,” a Sport In spokesman said.
The Stroytransgaz company, which built the stadium, had yet to issue a statement on the incident.