You MUST present a valid US passport to the gate/ticket agent before boarding. The passport is then scanned. Various computer readable information is on your picture page (and maybe inside the jacket for newer ones, thin RF tags), including your passport number. So I'd be pretty sure your passport number is known by the airlines at time of check in.

When we check in online, one of the input fields is the passport number.

More information can be found here: http://www.faasafety.gov/hottopics.aspx?id=47