About Switching employment in USA

I am working in a consultancy firm , for their client XXX.
XXX acknowledges me as its temporary employee, it provides training, defines job responsibilities, but does not pay me (termed as non-exempt employee).I am offered employment though the consulting firm and paid by consulting firm.

Earlier, before employment I had applied for xxx as part of my job search, I got a call for onsite interview ( i.e. after accepting my employment offer and working in consulting firm)

Glad, the interview went well and I am being offered employment. SO, now comes the problem. I am working in a consulting firm for a client xxx, which is offering me employment.

I need to know the possible issues / confrontations that could affect me from both the consulting firm and the client xxx

Thanks,
Tamm