This issue is not yet specifically addressed in the guidance from the IRS. If your child has a social security number and you have been claiming your child as your qualifying child on your tax return, you would receive your $1,200 (or other amount based on your income) plus the $500 for your child if the child is age 16 or younger.
If you have been filing a joint return with your spouse, including your spouse's income, and have made the election to treat your spouse as a resident alien - we have no information on how your wife would be treated but the law seems to indicate that taxpayers with an ITIN rather than a SSN would not qualify.
If you have been filing using the Married Filing Separately (MFS) status, it is believed that you would receive only the amount for yourself and your child, if your child has a social security number.
However, this is all tentative information and things are changing rapidly.
Check frequently for updates at each of these links:
Coronavirus and your Taxes: TurboTax.
Coronavirus and your taxes: IRS.
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