DT gives that the song is "Collected by Robert Burns."  Almost certainly this comes from Ewan MacColl's fine Songs of Robert Burns (Folkways, 1959).

But I had a look in Dick's The Songs of Robert Burns, and it does not appear.  While Dick (1902, with updates) is not exactly gospel, it is superb and has extensive notes.  Burns was certainly familiar with "I maun hae a wife" and used the tune for a last political satire, "Wha will by my troggin?"

Dick feels that the Northumbrian song was popular in the south of Scotland but that it does not appear in any Scottish collection, including printed Burns or Burns' manuscripts.

FYI, Dick goes on (following B & S) that the song is attributed, without evidence, to William Purvis ("Blind Willie") "an eccentric blind fiddler born in Newcastle in 1752."  It "was Willie's chef d'oeuvre in the streets and public houses that he frequented."