Looks like you're doing internship as a CPT and not OPT. If that's the case, the answer is NO. Once the duration of CPT (stated on your I-20 you got before starting internship) expires, you cannot continue to work. If you continue working, it'll be considered working illegally - just like working at a mall/gas station. If you are doing the internship just like working anywhere else (i.e. without going through the CPT approval process), it's already illegal and working for a few more weeks/months shouldn't do much more harm, if any.

Regardless of when your internship ends, apply for OPT exactly 3 months before you graduate, when applying put the day after your graduation date as your OPT start date, get hired at the company as soon as you graduate, start working without any trouble.