I have a feeling that I'm about to be fired. A coworker believed I lied and I've been struggling with depression. Should I ask my manager, perhaps off record, if anything is going on? Thanks

What you should tell you manager you heard through the grapevine that someone reported something derogatory about you and that you would like to discuss it with him. Don't ask about termination

What is the alleged lie, generally speaking? Does the coworker have any authority? Depends on a lot of circumstances. You could be making it worse depending on how you present it to your boss and we don't have enough knowledge of your situation, or the personalities involved. Exactly what do you mean by off the record?
" A coworker believed I lied and I've been struggling with depression." The coworker believes you have been struggling with depression OR you are stating that you have, in fact, been struggling with depression?