Kozak will be replaced as regional development minister by Viktor Basargin, deputy presidential envoy to the Urals Federal District.
"I will be ready to sign several decrees in the near future," Medvedev said at a meeting with Kozak and Basargin.
The president said he had endorsed the government's proposals on Cabinet changes, which Prime Minister Vladimir Putin put forward during his meeting with the president earlier Tuesday.
Kozak, who will become the eighth deputy prime minister in the Russian government, is widely perceived as a close Putin ally.
A professional lawyer, he held positions in the St. Petersburg administration in the 1990s, including deputy governor. He also headed the Russian government staff in 1999.
Kozak held several important posts in the presidential administration in the early 2000s until he was appointed presidential envoy to the Southern Federal District. In 2007, Kozak was named minister of regional development.
Sochi won the right to host the 2014 Winter Olympics at an IOC session in Guatemala in July 2007 after a close race with South Korea's Pyeongchang and Austria's Salzburg.
Russia will be hosting the Winter Olympics for the first time. Moscow hosted the Summer Olympics in 1980, but the event was marred by a U.S.-led boycott involving more than 60 countries.