Unlikely.
Plugging numbers into the Stefan–Boltzmann law gives us a temperature near 273°K (0°centigrade) for bodies near earth's orbit. The exact answer for atmospherless bodies depends on albedo.
Any water on nearby asteroids will thus boil until it freezes, and then sublimate.
That's why the search for nearby ice is focused on lightless, cold regions of craters near the moon's south pole.
Space probe Rosetta and its attendant comet are still 200 million km from the sun, and already outgassing water. IIRC, that started back in January, when the comet was 390 million km from the sun, well beyond Mars. Black body temp. out there would be around -100°C.
Looks like I was wrong about when jets first appeared: More jets from Rosetta's comet! September 19, 2014. On that day, the comet was 500 million km from the sun. That's outer belt. No spectra that I know of, so possibly not water. Water seems most likely though.
Likely we'll have to go out at least that far to find ice on small bodies.
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