Putin asked for the outgoing government to remain at work until a new government was appointed.
"We should provide the president of our country with the possibility to take all the necessary measures" to carry out the changes, Medvedev said.
Under the current constitution, which sets a maximum of two successive terms, Putin is barred from immediately running again, but his supporters find it hard to imagine Russian political life without him.
Mr. Putin, on a roll overseas but becalmed at home by Russia's stagnant economy, has a long record of wrong-footing friends and foes, and he has ensured that he has remained the arbiter on all important decisions in Russian Federation since Mr. Yeltsin resigned on Dec 31, 1999. We agreed that since Dmitry Anatolyevich has always dealt with issues of defense and security, we should introduce the post of deputy head of the Security Council.
The parliament's deputy speaker, Alexander Zhukov told reporters after the speech, that the national vote on the amendments would likely take place this year, perhaps in September. He spent four years before that as president from 2008-2012.
"These changes do indicate a change in power structures, but at the same time, they do keep the president very strong", Godfroid judged.
Dmitri Trenin, head of the Moscow Carnegie Center, said Putin appeared to be moving to limit the power of a presidential successor.
In his address, Putin said the constitution must also specify the authority of the State Council consisting of regional governors and top federal officials.
"Putin himself would not like there to be a second Putin", he said.
The Belgian activist suggests this move will "help refine democracy in Russia" stressing that even in the West "not many state leaders dare" to make such a bold step. Medvedev then said Putin should be prime minister - moves that critics decried as an imposed job-swap without input from the electorate. His replacement Mishustin is not known as a major power broker or as from Putin's inner circle.
Mikhail Mishustin, 53, is the head of Russia's tax service, credited with bringing digital tools to revolutionise the agency and help it crack down on tax evasion. This is not what we hear normally in the press about "Putin's Russia" and his personal control.
Vladimir Putin's selection for Russia's next prime minister is a loyal technocrat seen as a capable placeholder while the Russian president plans for his political succession. "And in these conditions, I believe that it would be right, in accordance with Section 117 of the constitution", for the government to resign, Medvedev added. Putin himself treats United Russia like an unpleasant relative he is occasionally required to visit in the hospital despite the party's energetic support for his presidency.
The annual speech to top Russian officials and members of parliament largely focused on how to improve domestic living standards.