Baldwin filed suit amid allegations Costner persuaded him to sell his cut in Ocean Therapy Solutions, a firm which builds devices to clean up oil spills, shortly before company bosses landed a multi-million dollar deal with BP executives to help them clean up the Gulf of Mexico spill in 2010.
Alec Baldwin's brother claims Costner and his business partner Patrick Smith failed to share information about a meeting they had had with BP head Doug Suttles to put an $18 million (£11.25 million) deposit down on a $52 million (£32.5 million) order when he was convinced to sell up.
Costner argues there was no firm deal in place when he learned Baldwin was planning to dump his shares. The Flintstones: Viva Rock Vegas star is demanding $4 million (£2.7 million) in compensation.
Both actors were in court in New Orleans on Monday (04Jun12) when prospective jurors were screened for the case, but they did not interact.
Costner recently spoke out about the lawsuit, admitting he finds it ""frivolous"" and ""upsetting"", but he has vowed not to give up the fight.
Last month (May12), he told The Hollywood Reporter, ""In my mind, it's a frivolous lawsuit. But I won't stop, no matter what. We have (oil) spills every day around the world. We don't have an adult response to how to clean it up.
""I knew how to do it 20 years ago in a very engineered, scientific way. But the oil industry didn't beat a path to my door. Twenty million dollars of my own money later, I have the Baldwin lawsuit, and it's completely upsetting to me.""