Is second stimulus check counted as 2020 or 2021 tax year?

You will have to enter both stimulus payments. Please follow these steps:

1. Log into TurboTax.
2. After entering all your information click "Federal Review" at the top of your screen. 
3. On the screen "Let's make sure you got the right stimulus amount" click "Continue".
4. You will be asked about the amounts of first round ($1,200) and second round ($600) stimulus payments you received.  If you did not receive a payment enter "0".
5. TurboTax will then calculate the amount of stimulus payment remaining that you are still entitled to get. 
6. Any stimulus amount remaining due to you will show as a credit on line 30 of the 1040

Yes a payment you got in January or February 2021 is an Advance and based on your 2020 return.  Enter the second Stimulus payment on your 2020 return now.  The year you got it doesn't matter.  If you don't report it Turbo Tax will give it to you again, which would be wrong and double.  The Stimulus entry screen says the second one is between Dec 2020 and Jan 2021 so you should include any payments you received in January and February.

Both stimulus checks are for 2020. The checks are non taxable. If you hadn't received a stimulus or it was too little, the underpayment will be issued as a credit on your 2020 return based on your 2020 income tax return after entering the stimulus amounts received.

Is the second $600 stimulus check received on 3 January 2021 reported on 2021 Federal tax return or must the Federal 2020 tax return be amended.

You are not asked about the 2nd stimulus on a 2021 tax return.   You are only being asked if you received EIP #3--the one for $1400.   You get IRS letter 6475 for it.

If you are not sure what you received then check your bank account and your IRS account.

https://www.irs.gov/payments/your-online-account

The 2nd $600 goes on your 2020 return, not 2021.   But you don't need to amend 2020.   If you didn't include it then the Recovery rebate credit on line 30 might have been too much.  The IRS would have automatically fixed that and adjusted your refund for you.  They know they sent it to you.  

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@dburns1236 wrote:

Is the second $600 stimulus check received on 3 January 2021 reported on 2021 Federal tax return or must the Federal 2020 tax return be amended.

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The second stimulus payment of $600 was to be reported on your 2020 federal tax return.  If you filed the 2020 return without reporting the payment there is no need to amend since the IRS knows that they sent you the payment and would adjust the 2020 federal tax return accordingly.