This paper shows that discrete Morse-Bott theory can be developed as a natural extension of R. Forman's discrete Morse theory by improving the definition of the discrete Morse-Bott function introduced by S. Yaptieu. To this end, we demonstrate that the combinatorial structure of critical cells can be extended to critical sets intuitively. Furthermore, we establish the discrete Morse-Bott inequalities, providing a unified view that extends both the discrete Morse inequalities and the continuous Morse-Bott inequalities.