Negotiating placement start dates?

So, I managed to get an offer for a research summer internship in Canada, which I really want to accept. However, I also plan and am applying for placement years next year. Some start in September which should be fine and not clash with my internship, but some advertise their start dates as early as June.
So now I'm considering asking the research lab if I can complete a portion of my internship remote while starting a placement, if I happen to get one that starts earlier. Or should I instead try to negotiate later start times with the placement. I don't know how flexible industry is, so I don't know how likely it is they agree. I assume a research lab is more flexible, but my internship is being funded by a larger organisation, and I also don't want to ask for too much. Also I don't know if its even allowed to complete an internship while working full time.
Any advice?