The answer you need is in the instructions of F.1040, which says the following:

Married Filing Separately

Check the “Married filing separately” box at the top of Form 1040 or 1040-SR if you are married and file a separate return. Enter your spouse’s name in the entry space below the filing status checkboxes. Be sure to enter your spouse’s SSN or ITIN in the space for spouse’s SSN on Form 1040 or 1040-SR. If your spouse doesn’t have and isn’t required to have an SSN or ITIN, enter “NRA.”

If your client elects to file jointly, the NRA spouse will definitely need an ITIN if he/she is not eligible for an SSN.