Tang Tay-ping, president of CSBC Corp, Taiwan, told the Taipei-based Commercial Times that his company was expected to obtain the order before the end of this month.
If finalised, the contract would require construction of 10 8,000 teu container vessels at a total price of 33.1 billion Taiwan dollars, with the first ships due for delivery in 2014, the paper said.
Evergreen said the deal was not finalised when reached by AFP for comment.
The report came after Evergreen Marine, the world's fourth largest shipping firm, ordered 10 ships from Samsung Heavy Industries of South Korea for $1.03bn early this month.