Putin also proposed including a requirement that future presidential candidates must never have held a foreign passport and a declaration that Russia's constitution transcends global law, so the Russian government can ignore rulings from worldwide courts.
But exactly how the balance of power between the executive and legislative branches of government will change, and where Mr Putin himself will end up in that arrangement, remain as clear as mud.
President Putin's fourth term in office is due to end in 2024, and under the existing constitution he would not be entitled to another term.
BBC Moscow correspondent Sarah Rainsford said the reason why Mr. Putin had removed Mr. Medvedev was unclear.
The package of reforms would be put to a national vote, he said, without specifying when.
"In this context, it's obvious that we, as the government ... should provide the president of our country with the opportunity to make all the decisions necessary for this".
Medvedev's resignation followed Putin's annual state of the nation address earlier Wednesday. According to The Guardian, Medvedev, after stepping down as prime minister, was appointed to a new position as deputy chairman of Russia's Security Council.
When Mr. Medvedev held the presidency, he and his patron were not always in sync, to the annoyance of Mr. Putin, who undid several of his measures. The president now holds the authority to make those appointments.
Mishustin is credited with making the Russian tax service highly efficient and supports electronic government services for the public.
Putin replied, "We were able to see that you are a brilliant dancer", to which Bush responded, "We'll leave it at that".
"This will increase the role and significance of the country's parliament", Putin said.
The five permanent members of the United Nations Security Council (UNSC) should set an example to remove the prerequisites for new global wars, Putin added. Alexi Navalny, Russia's most outspoken opposition leader, tweeted that "remaining the sole leader for life, taking ownership of an entire country, and appropriating wealth to himself and his friends is the only goal of Putin and his regime".
During a marathon press conference in December, Putin also hinted that the constitution could be changed, suggesting the word "consecutive" could be removed from a constitutional clause that states no individual can serve beyond two consecutive presidential terms.
Speculation has swirled about changes to Russia's political system that would allow Putin to stay on after 2024.
Russia's population fell dramatically in the 1990s in the tough economic and social climate after the collapse of the Soviet Union, and Mr Putin has faced demographic problems for much of the time since he became president in 2000.
Russian Prime Minister Dmitry Medvedev at the 2016 Munich Security Conference at the Bayerischer Hof Hotel in Munich, Germany, on February 13, 2016.
He obediently stepped down after just one term as president and let Putin reclaim the top job in what was widely seen as cynical political maneuvering and triggered massive protests in Moscow. Russian Federation has struggled to improve performance on those in recent years amid low oil prices and western sanctions.