"Here we need a bridge to take both cars and trains," Putin said at a meeting with his ministers, cited on the Kremlin website.
The bridge would allow Russia to deliver people and cargoes directly to Crimea without going through mainland Ukraine. Currently there is only a basic car and pedestrian ferry service across the 4.5 kilometre wide Kerch strait.
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Ukrainian officials previously put the cost at $1-3 billion.
Sokolov told Putin that Russia was also studying the option of building a rail tunnel under the seabed, where he said there were "complex geological and hydrological conditions."
"We will carry out all these parallel processes as quickly as possible in order to start the construction of the bridge as soon as possible," the transport minister said.
A spokesman for state monopoly Russian Railways, cited by RIA Novosti news agency, said the company was "ready to take part in developing the project".
Early this month, Russian Prime Minister Dmitry Medvedev put federal road agency Rosavtodor in charge of the bridge project.
The project had been agreed in December between Russia and the former Ukrainian government of now ousted president Viktor Yanukovych.
A bridge between the Crimean town of Kerch and the long Chushka spit on the Taman peninsula in Russia's Krasnodar region has been envisioned for decades since the Soviet era.
The German army began to construct a link in 1943, when the Crimea was under Nazi occupation.
Soviet authorities later finished the bridge, but it was destroyed by shifting ice even before the end of World War II.
In the 1990s, a new bridge was lobbied for by then-mayor of Moscow Yury Luzhkov, with reports estimating the cost at about $1 billion at the time.
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