Here is the text in Belden:

BIRD IN A CAGE
1.
Say that you love me,
Give my heart ease!
As soon as my back is turned
Love who you please.
Chorus:
Bird in the cage, love (3X)
Singing so sweet.
2.
Willy, my darly,
My darly dear,
If you don't love
Foolish I fear.
3.
Build me a nest
In the tower so high
So I can see you
When you pass by.
4.
I'll write you a letter,
Some two or three lines;
When you answer
Say you'll be mine.
5.
My parents don't like you,
That you well know;
They say you are not worthy
To knock at my door.

"This somewhat incoherent song combines the idea of security in a high place with the quite different image of a captive bird. Sandburg A(merican)S(ongbag) prints a text and tune from Kentucky."

"From the MS miscellany of William Dresia of Columbus, Kansas, sent to me by Miss Lowry in 1909."
H. M. Belden, ed., 1940, "Ballads and Songs," pp. 488-489.

An untitled song with elements of the above, also with the general form of "Down on the Levee" or "Down in the Valley" or "Birmingham Jail," in Folksongs of Florida, has the verses about Willy, and build me a ..., but no bird in a cage.