Wales, surprise winners of the tournament in Dubai four years ago, will be in the same group as sevens giants Fiji, plus Tonga and Uruguay.
England, meanwhile, have been bracketed with Argentina, Portugal and Hong Kong, while Scotland are alongside hosts Russia, South Africa and Japan.
Probable tournament favourites New Zealand have landed Canada, the United States and Georgia following Thursday's draw at the Petroff Palace in Moscow.
The six pool winners and two best runners-up will secure quarter-final places.
In the 16-team women's World Cup, England are in Pool D alongside Russia, France and Japan, with Ireland facing Pool B competition from defending champions Australia, South Africa and China.
The tournament, a key stepping stone on the road to rugby sevens' appearance at the 2016 Rio Olympics, takes place from June 28-30.