The duma, or lower house of its federal assembly, passed the scheme, which is part of a $4bn project.
Russia annexed the former Ukrainian region last year and ferry operators have since seen traffic increase threefold across the Kerch Strait, business that a fixed span would jeopardise.
The scheme will be managed by Stroigazmontazh, the Moscow Times reported.
It must now be approved by the upper house.
The work could be completed by the end of 2018.
