John A. Allison IV built one of the largest U.S. regional banks in the years leading up to the financial crisis. Since then, he has remained a fierce champion of free markets and a vocal opponent of new government regulations intended to stave off the next meltdown.

Now, Mr. Allison may find himself with greater opportunity to influence policy. The 68-year-old North Carolina native met Monday with President-elect Donald Trump. In an interview Tuesday, Mr. Allison said he spent an hour to 90 minutes with Mr. Trump and others,...