For the verb meaning to charm or to inflame with love, enamor is the standard spelling in American English, and enamour is standard everywhere else. This extends to all derivatives, including enamored/enamoured and enamoring/enamouring.
Enamour is one of a large group of words containing an our syllable from which the u has been dropped in American English. It is somewhat interesting, though, that enamour appears to have been a holdout for a long time. The more American form, enamor, has existed since the 19th century, but, as a Google ngram illustrates, it only recently became the preferred form in American English. Most words that have dropped the u in the U.S. did so long ago.