I don't know if you're in the US, but here you are required to keep at least 4' of clearance between yourself and a bicyclist when passing them. If you can't pass safely while leaving that much of a cushion, then you don't pass them until you can do it safely and legally. This is even on the CDL test (I'm a school bus driver myself). Safety first, schedule second.

Personally, I don't pass cyclists unless I can safely go all the way into the other lane. 4' is really not enough space -- at that distance a cyclist could easily still fall under your wheels if they suddenly go down. That being said, for me the most dangerous situation is when cyclists ride on the wrong side of the road, against traffic. This is completely insane behavior but something I'm seeing more and more often where I live and work.