Visa Issues Complicating Interviews

Hi all, I am a foreign national and am currently based outside the US. I am interviewing with some top US based pe firms, and have already had one or two rounds with each. I was supposed to get my US citizenship next month after which I would have immidiately flown to US for hopefully final interviews. I have run into some unexpected administrative processing issues with US authorities and am expecting the process now to be delayed by around 4/5 months. Whatever happens, I am certain I will be able to move to the US by June.

My concern is how do I break this news to my interviewers who are expecting me in NY next month?? Do I just be candid (and disclose these visa delays) or do I make up a story of some mishap happening and request to do final interviews over the call....I am very keen on these roles and don’t want to lose them over issues that I know won’t exist in the next few months. I am even willing to pay the employer a certain amount to hold my spot if I am extended an offer. Please advise me as I am super stressed out. Thanks in advance!