It sounds like your blog is hosted on HubSpot and you wish to send blog notification emails through Mailchimp rather than through HubSpot's email tool. Is that correct?
No integrations I know of support this out of the box, so are you seeking to build a custom integration?
If so, I can certainly speak to how you might go about this from the HubSpot side, including: • Retrieving blog subscriber records via the Contact Lists API.