Based on your question, my presumption is that the agreement was executed before 2019 and your client is a US resident.

For purposes of claiming the deduction, ITIN/SSN is not technically required.  Not sure if you could override the critical diagnostic to e-file though.  If not, you may need to paper file the return.

Your client, however, is required by law to establish whether the NRA recipient is exempt from US tax with respect to the alimony received by domestic statues or tax treaty.  A W8-BEN would be required and an ITIN is necessary for withholding, remittance, and reporting.

If your client did not fulfill the fiduciary duty to withhold, remit, and report, he could be held liable for the actual tax that is otherwise due from the recipient and subject to various penalties.