For example if in answer to the question, "what time does the shop close?" a tourist information officer might say, "I'll check on that for you."
Why wouldn't they say, "I'll check that for you."?
Check means to examine (something) in order to determine its accuracy, quality, or condition. Check on is a phrasal verb that means verify, ascertain, or monitor the state or condition of.
It's a subtle difference between the two, and, in the example you gave, I think either could be applied.
If you asked me which one was more correct, I think I'd vote for check, rather than check on, but that was only after consulting a dictionary. Most tourist officers aren't that careful with their words before checking up on the hours of a village shop.
Oh, check up on means investigate in order to establish the truth about or accuracy of, which is, I suppose, another way of saying (nearly) the same thing.
Ref: these definitions were taken from NOAD
Check on is used to refer to verifying the legitimacy or the condition of someone or something, according to Free Dictionary.
I reckon, check is better suited in your example.
I think both are okay. For example if I worked at a hospital, I would use check on.
And if I were going through a security check, I would say:
But in your case, both could work unless you got very picky.
In Brit Eng, we would say check up on smbdy/smth) when talking about a state or process.
A bare check is reserved for facts and events. Hold on, I must check if I have turned off the gas.
Check on or check up on is better for processes and states, or for a more intensive search for information. My aunt is ill, I'll go and check (up) on her later. Or I want to phone the office to check (up) on how that difficult job is going.
I am not sure that the instant quote is a good example - a bare check is enough for me as the fact is easily ascertainable.