Without much detail, Putin thanked Medvedev and his government for their work, TASS reported.
Under the current constitution, the 67-year-old Mr. Putin is barred from running for the presidency again when his current term expires in 2024.
Mr. Putin described the proposed constitutional changes in his annual state of the nation address as an effort to enhance democracy.
Having announced the redistribution of power, Putin has stressed that Russian Federation still needs to remain a "strong presidential republic".
Following Medvedev's announcement, President Putin said he plans to offer him a new position as the deputy chairman of Russia's Security Council, a powerful consultative that shapes presidential decisions on national security.
Approval of the prime minister and other key cabinet posts, suggested Putin in his speech, should soon lie with Russia's Duma legislative body.
"These are very serious changes to the political system", said Putin in outlining the proposals. "Putin would like to have some leverage, some mechanism to control and to get involved in case his successor makes mistakes or has some disagreements with him".
Putin thanked Medvedev - who also served as Russian president for four years from 2008 - expressing "satisfaction with the results that have been achieved".
Medvedev will be deputy of the Security Council - out of the way?
Alexei Navalny, the most prominent Russian opposition leader, tweeted that the president's speech signalled Putin's desire to continue calling the shots after his term ends.
"But for these constitutional changes to be made, Putin needs support". But if that referendum will actually occur is unclear. Russia's Central Election Commission said Wednesday that a plebiscite could be organized "at any time, once the proposals are finalized". "He would not be allowed to reject candidates confirmed by parliament".
"Today in our society there is a clear demand for change", Putin said in his address.
In response to public discord, Putin spelled out social spending measures to counter Russia's declining population, which he characterized as a "direct threat" to the country's future, including a nationwide hot-school-meals program. He also touched on health care and environmental issues.
Wednesday's state-of-the-nation address was his first to be projected across several large buildings in Moscow.
In his speech, Putin also boasted about Russian weaponry. "We have to wait and see who it is", Stanovaya said.