My current tires are 700x23. I know I can switch to 700x25, because last year I spoke to someone at the LBS who had measured the frame for another customer who had the same bike, to see if she could put 700x25 tires on it. However I could not go wider than 25. I think the issue (and what he measured) is the amount of space inside the forks. I would not need new wheels to make the switch.

It may be possible for you to use a wider tire than what you have now, just not as wide as the ones you asked about. And you would not necessarily need different wheels.