Warm kitty, soft kitty,
Little Ball of fur,
Sleepy kitty, happy kitty,
Purr, purr, purr.
Popularised in recent years by TV comedy The Big Bang Theory, where the lyric is slightly amended, Warm Kitty was written by Edith Newlin and copyrighted in 1937. Sources differ on the origin of the accompanying tune, with some suggesting it is an English, and some a Polish or Hungarian folk song. I cannot find the folk song the tune is supposed to be based on, so if you happen to know it please let me know.
Intended as a calming and soothing song for quiet time, the piece was first published in Songs for the Nursery School, edited by Laura Pendleton McCartney. Thanks to the National Library of Australia, the book is available to view online.
The Big Bang Theory amends the lyric slightly, transposing ‘warm’ and ‘soft’ in the first line and ‘happy’ and ‘sleepy’ in the third. Whether this is down to the faulty memory of a writer or a deliberate change remains unknown.
The original version of the song, sung by Australian TV presenter Patsy Biscoe who recorded the song for her album 50 Favourite Nursery Rhymes Vol.3.

You are incredible!
Well, not really, but thank you! Have some kitten love.
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The tune is very similar to “Lightly Row” with only a few rhymic changes, and
“Lightly Row” has a short B section followed by a repeat of “Warm Kitty” here’s a link to the Suzuki version http://musescore.com/user/1977/scores/1994
and a version I put together that shows the parallels
http://www.noteflight.com/scores/view/3d1dd56e45b0f03f26cd6333c745cd4642bded80
whoops! Meant to say “rythmic”