There is not a way to claim a dependent without a Social Security number or ITIN (Individual Taxpayer Identification Number).
If you do not have a required SSN by the filing due date, you can file Form 4868, Application for Automatic Extension of Time To File U.S. Individual Income Tax Return, for an extension of time to file.
This IRS Publication states:
You must show the social security number (SSN) of any dependent you list in the Dependents section of your Form 1040 or 1040-SR. If you don't show the dependent's SSN when required or if you show an incorrect SSN, certain tax benefits may be disallowed.
No SSN
If a person whom you expect to claim as a dependent on your return does not have an SSN, either you or that person should apply for an SSN as soon as possible by filing Form SS-5, Application for a Social Security Card, with the Social Security Administration (SSA).
You can get Form SS-5 online at SSA.gov/ forms/ss-5.pdf or at your local SSA office. It usually takes about 2 weeks to get an SSN once the SSA has all the information it needs. If you do not have a required SSN by the filing due date, you can file Form 4868, Application for Automatic Extension of Time To File U.S. Individual Income Tax Return, for an extension of time to file.
Are you saying that the child does not have a Social Security number? Or does your child have a Social Security number but the name is incorrect? If the child has a Social Security number but an incorrect name, you may claim the dependent under the name on the Social Security card.
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