Integration with is the commonly accepted term in professional usage.
To see this, go to the Software Engineering Institute webpage at Carnegie Mellon University. Search for "integration-with" and again for "integration-to" (the hyphens seem to make their search engine look for adjacent words as well as hyphenated words).
That said, "integration to" is not completely wrong, since it could convey the idea that a small thing is being integrated into a larger thing at a more remote location, as in a software component being deployed on a mobile phone.
BTW, being a native speaker is not the same as having domain knowledge and a professional vocabulary. However, if your manager is technically competent and you feel comfortable asking, you might find that their choice of words is conveying an understanding of the system that could be useful for you to know.