to inspire (inspires, inspired, inspiring)
1. To breathe into; to fill with the breath; to animate.
2. To infuse by breathing, or as if by breathing.
3. To draw in by the operation of breathing; to inhale; -- opposed to expire.
4. To infuse into the mind; to communicate to the spirit; to convey, as by a divine or supernatural influence; to disclose preternaturally; to produce in, as by inspiration.
5. To infuse into; to affect, as with a superior or supernatural influence; to fill with what animates, enlivens, or exalts; to communicate inspiration to; as, to inspire a child with sentiments of virtue.
aspire
1. to hope or dream; especially to hope or work towards a profession or occupation
Aspire is a intransative verb (an action verb not taking a direct object. Although there is some dispute over whether or not a linking verb is intransitive, many do not consider it an intransitive verb), which I presume is what your talking about.
Usually I would agree, however, I was going for the "to affect" meaning, not to "hope or dream".