The firm is one of a handful of companies who have contacted administrators from Grant Thornton about buying parts of the business.
A European and Asian firm are also said to be considering an offer for the remaining assets of the Welsh steelwork contractor.
Most of Rowecord’s staff have been laid off with around 70 kept on to see out the firm’s remaining contracts.
Rowecord is currently working on major projects at the Dover Docks and the British Museum.
A source told the Enquirer: “The word on the street is that Bourne are looking at making a serious offer for Rowecord.
“A deal could give the firm the critical mass to be a major player in the market.”
The steelwork industry has been through three torrid years but recently firms have reported signs of improvement in the market.
Competition for work is expected to ease after market leader Severfield Rowen announced that it would be more cautious about bidding for future work after losing £20m on several London jobs.
Bourne has been keen to raise its standing to bid for top-tier jobs and last year formed a joint venture with rival Billington to target high-rise work in London.