Its Appledore shipyard in Devon would construct the vessel in time for its first mission in 2019 if successful.
The ship, to be operated by the Natural Environment Research Council, will be one of the biggest research units in the world.
The 90-metre, 2,256-gt vessel will be able to work at both poles.
A spokesman for Babcock told local media: "Babcock are planning to prepare a submission as part of an open and competitive EU tendering process for a new polar research vessel."
