How can I predict if the binding site of a DNA-binding protein is located on the promoter sequence of a gene? Is there any computational tool or any other method to predict if the binding site of DNA binding protein is actually located on a promoter sequence?
You could run FIMO on your entire genome for TFs (transcription factors; DNA-binding proteins) of interest, which gives you predicted binding sites: genomic intervals where those TFs bind, with some statistical significance.

https://bioinformatics.stackexchange.com/a/2491/776

You can then map or intersect promoter regions with those TF binding sites with BEDOPS bedmap or bedops, respectively:

https://bedops.readthedocs.io/en/latest/content/reference/statistics/bedmap.html

https://bedops.readthedocs.io/en/latest/content/reference/set-operations/bedops.html

You can define your promoter regions however you like, e.g., from the TSS to 2.5kb upstream of the TSS. It's up to you and your experiment.

The use of bedmap is recommended to assign TFs to a promoter of interest. This tool will return both the promoter region and any TFs that overlap it by the specified overlap criteria.