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If I was to negotiate salary for a job offer with HR, is it customary for HR to communicate this to the hiring manager?

I'm worried that it colors my relationship with the hiring manager if I accept the offer. It's an indicator of how much I want the job ("I'll work for you, but not for what you wanted to give me."). Depending on what I'm negotiating for (salary in this case, but in other cases leave, benefits, etc.) it communicates work-life priorities. All subtle but important communicators that change my relationship with the manager before I even start.

As answered, yes, you should assume the hiring manager is / will be aware.

In response to what you said in the comment:

It colors my relationship with the hiring manager if I accept the offer. It's an indicator of how much I want the job ("I'll work for you, bit not for what you wanted to give me.").

Most managers understand that you are accepting a job in a large part because you want or need money, and will not hold this against you. Additionally, everything that could be interpreted negatively can also be interpreted positively. For example, asking for more money can demonstrate that you have confidence in your own value.

When you ask for raises after you've accepted the job, that will also start with your manager. Are you going to hold back then so your manager doesn't think you're greedy? If you aren't willing to advocate for pay raises in your career, then you will end up underpaid. Lastly, I've never had a manager hold looking for a raise against me. Work is a business transaction and most people understand that. As long as you aren't jumping jobs every year and expecting a huge raise each time you'll probably be fine.

Yes this is normal and you want that to improve your chances.

HR normally has little latitude in negotiating salaries on their own. They are not normally qualified to evaluate whether a particular candidate is worth more than was offered. The hiring manager should be aware of what you are worth to fill the role and whether or not they are willing to go higher. If the company allows it at all, it is the hiring manager who is quite likely the person who can approve or initiate a higher salary offer.

If I was to negotiate salary for a job offer with HR, is it customary for HR to communicate this to the hiring manager?

Yes this is a normal hiring procedure.

Yes, it is normal for HR to communicate negotiations with the manager you will be reporting to.

One thing to consider is that the manager has a budget so they need to be aware of things that would change their budget.

Your idea what your manager thinks about your salary is totally upside down. If your salary is low, your manager will think "that person can't be as good as I thought, or they would have negotiated a higher salary. If your salary is high, your manager will think "HR must have seen that this person is worth more money", or "this person must have been good enough to get high offers elsewhere, or HR wouldn't be paying that much".