Let's start with a few facts.
First, whether you're getting a paper check or a direct electronic deposit, any COVID Relief funds from the federal government are being handled by the IRS and the U.S. Treasury. Not the Federal Trade Commission (FTC).
Second, regardless of the alleged reason, the FTC is not going to send you an email message out of the blue. Specifically, the agency's Acting Chair Rebecca Slaughter is not going to send you an email message, regarding COVID Relief checks or any other subject, requesting personal banking information.
However, there is a recent crop of email scams that ignore these facts and attempt to harvest personal information and account details from potential victims. The messages spoof Rebecca Slaughter's email address at the FTC, and while I have not seen the full text of one, I would bet on some variation of "we need you to verify your information to get your relief payment."
Federal agencies generally do not initiate contact with citizens by email, nor do they ask for personal identifying information. Since there really is no such thing as an "official" email address for each person, it would be impossible for them to know which one to use at any given time—personal email addresses can be terminated or abandoned and new ones created instantly, work emails can end suddenly if the owner's employment status changes. If the IRS or Social Security Administration (for example) really need to contact you, they will primarily use postal mail.
That said, you can sign up for Consumer Alerts from the FTC that will help keep you informed of different types of scams. You can sign up for those here, and you will get emails from the FTC in this case. However, notice that YOU are the one initiating the contact, not the FTC, and they won't be asking you for account or other personal information.
