John Fredriksen's Flex LNG wants to move its listing up to the Oslo Bors from the Oslo Axess board.
The company has submitted an application to the stock exchange, it said on Tuesday.
Oslo Bors represents a full stock exchange listing that complies with all EU requirements. It is favoured by larger companies.
Axess is suitable for companies that have less than three years' history. It has less detailed requirements.
Clarksons lists Flex LNG as having six LNGCs on order at DSME and Samsung for delivery in 2018 and 2019.
In April, it was reported that the company was carrying out a private placement of shares aimed at raising NOK 1.1bn ($125m) to fund the ships.
