More is needed than just a suggestion. A suggestion by the supervisor to the employee to use the EAP is not a supervisor referral. Don't confuse these discussions with your employee as past attempts to make a supervisor referral. Instead, a supervisor referral is a direct action step--one that gets the employee's attention and motivates corrective action.
Avoid a misstep. The misstep is moving directly to disciplinary action thinking past discussions with the employee about the EAP were ignored. The consequence is the potential loss of an employee who may have responded promptly to a formal supervisor referral.
They don't work well. Unfortunately, the usual outcome of informal discussions about the EAP and the hope that an employee "will take a hint" is an employee who makes more attempts to control symptoms. These employees often get worse, and the relationship with the supervisor does too.
It's not the same as an "informal referral." Your organization may define an "informal supervisor referral". Usually these are direct supervisor referrals based on factors that do not warrant the threat of disciplinary action. They may include observable signs and symptoms of a personal problem that affects the workplace or the employee's disclosure of personal problems.