The proposal comes in a bill that MPs are expected to approve. The parliament is dominated by MPs loyal to Mr Putin.
The National Guard, launched last year, is headed by Mr Putin's ex-bodyguard Viktor Zolotov, a longstanding ally.
Mr Putin has appointed 11 new governors ahead of a March presidential election. He is expected to seek a fourth term.
The new men are overwhelmingly younger than the governors they replaced.
Russian governors are appointed by the Kremlin and generally come from the Moscow power structure. Some have little previous connection with the region they are directed to manage.
Currently elite state bodyguards are assigned only to the president, prime minister, top parliamentary and justice officials and visiting foreign leaders.
The National Guard's main tasks are: combating terrorism, protecting vital installations such as nuclear power stations, guarding arms depots, preventing public disorder and stopping the black market trade in arms and drugs.
The force's chain of command leads directly to the Kremlin, bypassing the defence and interior ministries.
Mr Putin's bill also calls for the National Guard to protect unspecified "other figures", besides the governors.
Mr Putin has previously warned that Russia must protect itself from "colour revolutions" like those that toppled presidents in neighbouring Ukraine, Georgia and Kyrgyzstan.
The National Guard - known as "Rosgvardiya" in Russian - has more than 340,000 personnel, about 160,000 of whom come from the interior ministry forces, Tass news agency reports.
The force has some of Russia's most sophisticated military hardware and crack troops from special forces units. Some are specialist divers and parachutists.
Deploying the highly-trained troops across Russia's 89 regions and republics could enable the Kremlin to quickly quell any election-related unrest.
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