Hi Psb,
Is't in Kualalumpur? If yes, I don't the practice in Malaysia. But I'm giving this views based on Indian conditions only.
As far as I know, Internship is Not a job per se. There may not 'employer-employee' relationship existed yet. It's only part of your curriculum. Students are doing part-time job, like wise this. Many of the courses comprised of compulsory Internship of prescribed period, which I hope you have completed by June end and received the completion certificate also. Few days this side shouldn't a problem. So long as the hiring Co have no problem is accepting your Internship certificate completed by 30.6. you'll have no problem. My opinion is you shouldn't hide the fact, especially when they asked you to produce this certificate you'll have no option either. Simply show the xerox of the certificate to the concerned HR in advance before submission 'without highlighting the overlapping the date' on your own. Only clarify position if you are asked to. See what transpires then act depending on the development. My hunch is you'll have no problem on this issue.