The following was set to music, and I saw it in a collection (not of "children's songs) in the 1980s. I have tried Worldcat, but cannot find anything there. thank you. Sanjay Sircar

In Laura E Richards, In My Nursery. Boston: Little, Brown © 1890, p. 88 (originally in St Nicholas, frequently anthologised):

AN OLD RAT'S TALE.

He was a rat, and she was a rat,
And down in one hole they did dwell.
And each was as black as your Sunday hat,
And they loved one another well.

He had a tail, and she had a tail;
Both long and curling and fine.
And each said, "My love's is the finest tail
In the world, excepting mine!"

He smelt the cheese, and she smelt the cheese,
And they both pronounced it good;
And both remarked it would greatly add
To the charms of their daily food.

So he ventured out and she ventured out;
And I saw them go with pain.
But what them befell I never can tell,
For they never came back again.