You can claim a non-citizen child as a dependent if they meet the IRS definition of a "qualifying child" and have a Social Security number or ITIN.
https://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-credits-deductions/claim-dependent/L...
An ITIN, or Individual Taxpayer Identification Number, is a tax processing number only available for certain nonresident and resident aliens, their spouses, and dependents who cannot get a Social Security Number (SSN). It is a 9-digit number, beginning with the number "9", formatted like an SSN (NNN-NN-NNNN).
https://www.irs.gov/taxtopics/tc857
You would need to apply for the ITIN with form W-7. Along with the completed Form W-7, you'll submit identity documents and either a federal tax return or other documentation to show the federal tax purpose for which you need the ITIN.
I am unsure as to why the IRS said that your dependent cannot have an ITIN.
@GuyBoren Thanks for the question!!
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