Checking accounts receive a lot of scrutiny in the US for various reasons, so it's preferable to keep them at separate banks if you plan on doing anything remotely boundary pushing. A checking account shut down will lead to a credit card shut down at most banks, including Chase.
If you are doing absolutely nothing that would raise an eyebrow, you likely have nothing to worry about. As always, YMMV and there are exceptions. Eg Capital One doesn't love foreigners.
