Whether your current (or any clincher) tires will fit on the new wheels depend on whether the wheels are tubular (sew-ups to some) or clincher wheels. It's pretty obvious, tubular wheels have no lip for a tire bead to grab onto. It'll just be a u-shape recess in the rim where you glue a tubular.

Assuming we're talking about a clincher rim here, you'll need to know two things:

1) With your current 700x25 (25-622 as fietsbob correctly pointed out) make a thorough visual inspection of the clearances that you currently have with that tire. If it's tight between the chainstays, the seat tube, the brake calipers or the fork crown, you may have an issue with a larger volume tire. If you have ample space in all of these areas, you've cleared the 1st hurdle.

2) Find out the ETRTO width of the rim. My guess is that if it's a 15mm rim. If so, the widest tire that you can put on and have no worries is a 700x32. If they are 13mm rims (and they may be if they are track rims) you are already at the max with a 700x25.