The constitutional changes, which Mr. Putin said should be put to a popular vote, would be the first major overhaul of Russia’s political order since 1993, when the country’s first democratically elected president, Boris N. Yeltsin, subdued a rebellious legislature with tanks and then ordered a referendum to endorse a new Constitution strengthening presidential power.

Unlike Mr. Yeltsin, who was deeply unpopular and faced an uphill struggle to get his new Constitution approved by voters, Mr. Putin has amassed such overwhelming personal power and popular support that he can be confident of reshaping the system in virtually any way he wants.

His popularity, reinforced by the Kremlin’s tight grip on television and many other news media outlets, allows him to add a veneer of democratic legitimacy and avoid the path taken by China, where the Communist Party leader, Xi Jinping, has effectively established himself as leader-for-life through executive fiat.

One scenario now considered likely is for Mr. Putin to leave his post as president and create a system similar to that of Kazakhstan. In that Central Asian country, the longtime president, Nursultan A. Nazarbayev, stepped down last year as his country’s formal leader but stayed on as the head of the ruling party and took the new title of “leader of the people.”

This has established Mr. Nazarbayev as the equivalent of China’s former paramount leader, Deng Xiaoping, who had no formal executive position in his later years but remained in overall control of his country until his death in 1997.

It was not immediately clear whether the resignations of Mr. Medvedev and his cabinet signaled a rift at the top of Russia’s hierarchy or — a far more likely possibility, according to most observers — were part of a well-coordinated but as-yet-unclear plan by Mr. Putin to hold onto power by reshaping the political system.