Comments

A longer version:

A flea and a fly
Flew up in a flue.
Said the flea to the fly,
'Oh what shall we do?'
'Let us flee,' said the fly;
'Let us fly,' said the flea;
So they flew through
A flaw in the flue.

Here's a version from "The Independent," Volume 59 (1905):

Flea, Fly, Flue

A flea and a fly in a flue,
Were imprisoned;
Now what could they do?
Said the fly, "Let us flee."
"Let us fly," said the flea,
And they flew through
A flaw in the flue.

Here are two versions about the flea and the fly having the flu (they seem to be old public service messages):

The first one comes from the "Monthly Bulletin Louisiana Board of Health," Volumes 8-17 (1918):

A flea and a fly feared the flu,
Which made them exceedingly blue.
Said the flea to the fly,
"I am sure I shall die."
Said the fly to the flea,
"I'm as scared as can be."
Said the flea, "Let us fly."
Said the fly, "Let us flee."
But they could not
Fly far from the flu.

For so quickly it flew
It infected the fly,
Who sneezed "Ah-Kerchoo"
And infected the flea.
Said the flea, "Death is nigh."
Said the fly, "Ah- Good-bye."
So they died and then
Flew up the flue.

The following one is from "F.A.C.O.T.S.: The Story of the Field Artillery Central Officers" (1919):

Said the Fly to the Flea,
"I've got the flu,"
Said the Flea to the Fly,
"What'll you do?"
Said the Fly to the Flea,
"I'll fly with you "
So the Fly flew away
Now the Flea has the flu.

Here's a similar rhyme from "Henry's Junior Dictation Lessons (1886) by Joseph Fernandez:

The little boy said
That the flea flew up
The chimney fine
But as fleas jump,
And do not fly,
He might flee,
But could not have flown.

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Recorded by Mama Lisa.

Thanks and Acknowledgements

Thanks to Monique Palomares for the illustration!