This song (of course, only a part of it) seems to be based on the rhyme (from the Levy collection):

Title: I'll Make a Ring Around Rosie. Song.
Composer, Lyricist, Arranger: Words by William Jerome. Music by Jean Schwartz.
Jean Schwartz Publication: New York: Jerome H. Remick & Co., 1910.
Form of Composition: strophic with chorus
Instrumentation: piano and voice
First Line: Rosie, Rosie, she thinks a lot of me
First Line of Chorus: I'll make a ring around Rosie, I'll make a ring around Rose