Learn English – A job interview “with” or “at”

Both of them are syntactically correct, and I've heard both. However, there can be a slight amount of nuance when using with versus at. This is mostly due to cooperation and how much you feel like you're "on the same team" as the company. Just as you work with your colleagues, you can interview with a company. Depending on the magnitude of the job, saying that you're interviewing with a recognizable company by name might convey more importance than at.

• I'm interviewing with Google: Implies that you're working to get a job at Google, a big company. It's identifiable and carries a sort of affection, as if you're already with them. I've heard this used primarily in the context of starting a long career or being at a very respectable or recognizable company.
• I'm interviewing at McDonald's: Implies that you see yourself working at the location, but not that you necessarily are close to the company. I've heard this more common when referring to jobs that aren't the end goal in a person's life. These jobs are usually "just a job" and a way to earn a paycheck. A stepping stone, not a final resting point.

In your example, you said "a clothing company," which, intentionally or not, removes some of the closeness in your context. If you were saying this verbatim, I would go with at (I'm going to have a job interview at a clothing company today) because you haven't given the name of the company directly. It doesn't give the closeness that with gives, but you've already distanced yourself slightly from the company by not giving its proper name.

I've also heard at used more in the context of a failed or disliked interview in past tense. For example:

• I interviewed at Google, and I didn't make it past the second interview.
• I interviewed at McDonald's, but the conditions didn't really give me the impression that I should work there.

Since the job wasn't attained, the implied closeness isn't needed. Many people want to distance themselves from unsuccessful attempts or careers that didn't speak to them.

All of this being said, however, I'd like to reiterate: both are technically correct and commonly heard. However, if you're trying to convey a tiny bit more information, you can do so by intentionally adding distance or removing it.