"Gentle Annie" has been compared with an Irish song (Wikipedia).
A broadside, unattributed, with Foster's lyrics was published in Glasgow in the 1850s by James Lindsay; it has a woodcut, and is included in the National Library of Scotland Digital Collection.
http://digital.nls.uk/broadsides/broadside.cfm/id/14824

In the commentary with the broadside, they state that the song is "now considered one of Foster's six finest songs." No comment is made on tune comparisons.