Did your wife live with you at any point during the last 6 months of 2023? If so, then no, you cannot file as Head of Household.
You would be able to file as Married Filing Separately. Since they do not have an SSN or ITIN and you choose to file Married Filing Separate ,you can try typing NRA in as your spouses last name. Make sure you put a check mark in the box that says My spouse is a Non-Resident alien in the My info section. When you get to the end and try to transmit the return, ignore the e-file errors regarding your NRA spouse and select transmit your return. If that does not work you will need to print and mail your return and write NRA (Non Resident Alien) in the space for her SSN.
If you file and the SSN is denied, the IRS would make adjustments to your return accordingly denying any credits or deductions that are not allowed for someone without a SSN.
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