You really think there's a number, or even a website that would let you do this? The potential for misuse through identity fraud would be tremendous. Right now, verification means that the employer selling out Americans certifies that they've seen some crappy piece of paper (whether they've seen it or not) that Jose Doe bought from some street vendor in Mexico. By the time the INS gets around to the employer, the crop is already harvested, and Jose, or whoever the hell he is, is down the road at some other farm.
People used the same thing on temporary jobs to keep from paying income taxes during the 1980's when I was a tax accountant. None of my clients did it, but I heard stories about workers claiming 99 exemptions, leaving their paychecks free of income tax withholding, and by the time the IRS told the employer to bust it back to "single, one exemption", the worker was already off the jobsite. The IRS figured out how to deal with this, why can't the INS?
The employer send that info into the govt every payday. It is the govt's responsibility to enforce the laws.