I got a job offer in a reputable company in USA and am going through background check.
Because you're a candidate for an employer in the USA, the employer must follow US employment law when it comes to background checks and how they impact the selection process. Most "criminal background checks" performed in the US are done via consumer reporting bureaus, which means they must comply with the Fair Credit Reporting Act, or FCRA. Per regulation, this means that the employer must:
- Get your permission to perform the background check. It sounds like this has already happened.
- Explicitly notify you that the reason you're not being hired is because of information in the report they received.
- Provide a copy of the report to you upon request, if the report is being considered as a reason to take a negative action (i.e. not hire you because of information in the report)
Regardless of how the employer obtained the information, they must be able to prove that they're not discriminating against you, as someone (theoretically) with a criminal record, unless the criminal record provides substantial evidence that you're not fit for the job. In other words, if your criminal record is totally unrelated to your job, and/or doesn't show a significant issue with you being a responsible and safe employee, they can't use it against you.
There are special cases where an employer may obtain (or try to obtain) criminal history via means that are not covered under the FCRA, but unless you're in a very specialized circumstance or applying for very specialized positions, that almost certainly doesn't apply to you.
For more reading on how employers are allowed to use background checks in the US, you can check the EEOC's website (although for the sake of keeping this a self-contained answer, I've summarized the pertinent info in the bullets above):
You're in a bit of a special case as you're an international candidate, so your employer may (or, apparently, is) use a specialized international background checking service, which many CRAs do offer. The method by which these services obtain data varies considerably from agency to agency and from country to country, but regardless of how the information is obtained, your employer must follow the law when it comes to applying it.
To bring this back around to your specific question,
What my options for what I could do next?
Right now, there's not much you can do, except seek out information on your own as suggested in @PlayerOne's answer - it would be worthwhile to understand your (potential) criminal record in your home country, or lack thereof. Also, it would make sense to have on hand the (positive) results of prior background checks if you still have them, from other jobs you've applied to.
Keep in mind that your potential employer's desire is to fill the position with a viable candidate. They're not in the business of trying to prove you did something wrong, they just want to know the truth about how you'll perform for them. Employers with experience will know that sometimes things come up wrong, especially on something like an international background check. If you have done your homework and can prove you have a clean history, that will likely make a good impact. If the employer still declines to hire you, and you feel they're not complying with the EEOC or FCRA or other relevant law, you can speak with an employment lawyer and determine what options you have.