What if your current salary is lower than market? I am concerned

hesitate to offer me the job if he knows how much lower I am making right now.  Will he think I am not competent? I am in a big Top Fortune 500 company, but the pay has not been competitive. My current company keeps employees by providing good benefits, flexiable schedule, work from home option, etc. I am shooting for an opportunity that will pay almost $30K more. Is that a too big jump? Will there be a problem when they find out how much I am making right now?