Can a potential employer obtain my employee record from my current employer?

I am currently working at a job and am looking for employment near home. My potential new employer has initiated a background check through an outside company with my permission. I see that they verify employment history but I want to know can that background check allow them to access my current employee record as well? I have warnings from over two years ago and a suspension that occurred due to those warnings. There has been over a year that has transpired since that suspension and no further warnings but I am concerned that the potential employer may be able to access this and even though they have offered me the position and everything is set for me to be hired they could decide to not hire me still after this background check is completed. That is the only issue in my background that would even be a red flag since I have no criminal record and an excellent credit score. I still work for the employer that is had the suspension with and the suspension was only for one weekend of work. I just want the Florida law regarding giving employee records or information in those records to potential employers.