@jp It is a violation of law to pass on the fees associated with H1B to the employee. The employer has to bear the cost of the fee that USCIS (the visa issuing agency) and a law firm (that preps the documents, files for LCA etc.) will charge. the ONLY fee that an employee can be expected to pay during an H1B is:
1) The roughly 2500 USD fee to expedite the application so that USCIS processes with a approve/deny within 15 calendar days. However, many big firms like a Citi or JP can easily absorb this
2)The dependent H4 where applicable but again, most firms pick up the tab and file jointly
3)Dependent work authorization if the petitioner is entitled to file one. Some firms will pick up the tab as a courtesy but most won't. This can also be self filed easily enough
Lastly, when Citi or any other employee says 'sponsorship not available' or 'we will not sponsor for this position' or similar verbiage, what they're saying is that the position is not open to visa holders so effectively only those with a green card or citizenship or work authorization via a program such as asylum/TPS etc. will be selected. Anyone with a visa will be rejected by the firm advertising a position with this kind of a verbiage as part of the JD