Worried about current employer finding out I am interviewing for other position

All,

I have worked at a small investment bank for two years as a summer intern, and I have a great relationship with my team/boss. I spoke to HR and they are sending me an offer this week when my internship. But, I am interviewing for other places and was told with someone I am networking with that I should let them know as soon as I get the return offer so they can fast track me. However, I am worried that the prospective employer will contact my current employer and that, in the worst case, they will rescind. 

My question is: Can HR really call your current employer as a reference/background check? I feel like they wouldn't do that because that could negatively effect you at your current job, especially if you have not let others know you are looking elsewhere to work. On the other hand, I feel like they wouldn't ask you to provide references for no reason. 

Thanks