Hi bringachange,
Raising a 560 to a 740 will require some serious work - and you'll have to make significant improvements to how you handle BOTH the Quant and Verbal sections. Learning, practicing and mastering all of the necessary content and Tactics will take time though - and 2 months will probably not be enough time. In all likelihood, you would need at least 3 months of consistent, guided study to get to that level. That schedule would take you through the end of August (or later), so if you were planning to apply for Round 1, then you would likely have to work on your applications while you were still studying for the GMAT. There's nothing inherently 'wrong' with that though, so this whole process now comes down to being efficient with everything that needs to get done. You have to keep your REAL goal in mind - to get into your first-choice Business School. It's likely that you don't actually need a 740+ to do that; it's also possible that you could apply for Round 2 and be just fine.
If you have any additional questions, then you can feel free to contact me directly.
GMAT assassins aren't born, they're made,
Rich