[Rick Lee]

If you want to roll the dice, you can call or write and tell them you have another offer forthcoming with a deadline for an answer and you're going to have to go with the bird in hand. This happened to my wife a few years back. She had an offer in hand for a place that wouldn't sponsor her H1 visa, but was pretty close to an offer for a job she wanted a lot more that would sponsor her H1. The second job wouldn't speed up their process other than to say they would be making her an offer, but couldn't say when. She ended up working the other job for about a month and then quit to take the better one.