If you can organise them to start _paying_ you before you start the job, then you'll really be set :D
That's quite common to not check references till someone actually gets the job. Checking references can be very time consuming, so why call up everyone's referees, when they can just check the successful applicants?
If your references don't check out, they'll let you know soon enough, and move on to Person B. That, or they weren't going to bother checking them in the first place.
And good luck for Monday.