Police were called to the accident on Burnley Road at 5.09pm yesterday (Friday).
The man was pronounced dead at the scene.
The bus driver was arrested on suspicion of drug driving. He has been released under investigation.
Police are appealing for anyone who saw what happened or may have dashcam footage to get in touch.
Information can be given via www.westyorkshire.police.uk/contactus quoting log 1282 of 3 June or by calling 101.
Burnley Road was shut both ways at its junction with Tuel Lane for several hours last night. It is now open again.