Enter the interest in TurboTax as if you did receive a Form 1099-INT. All you have to enter is the name of the bank, and the amount of interest in box 1.
You might still get a Form 1099-INT. The bank has until January 31 to put it in the mail to you. If the interest is less than $10 they are not required to send a form, but you still have to report the interest.