I don't know of any direct source. It's just the spirit of them nursery rhymes...

There is no "кошки-мышки" rhyme. "Играть в кошки-мышки" is rather an idiom used to describe a situation of trying to be evasive, usually figuratively, by trying to misrepresent, bend, distort and, sometimes, hide the facts from some sort of an authority.

Here's the cat, and here's the mouse,
Here's the camp, and here's the guardhouse...

I remember this poem (ПРЕДСТАВЛЕНИЕ) very well. Here's the whole thing:

http://www.lib.ru/BRODSKIJ/present.txt

Notice a bunch of these throughout the poem:

"Жизнь - она как лотерея".
"Вышла замуж за еврея".
"Довели страну до ручки"
"Дай червонец до получки".

"Не смотри в глаза мне, дева:
все равно пойдешь налево".
"У попа была собака".
"Оба умерли от рака".

"Сочетался с нею браком".
"Все равно поставлю раком".
"Эх, Цусима-Хиросима!
Жить совсем невыносимо".

Etc.

Occasionally, you will find references to known nursery rhymes such as "У попа была собака", but mostly, it's just Brodsky as if jeering at them, which becomes quite clear once you read the entire poem. Many of them just "reek" of Маяковский and Михалков in structure and rhyme patterns, and the trick is the new oh-so different content that is now occupying these oh-so familiar rhymes.

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Note added at 8 hrs (2007-07-26 02:51:03 GMT)
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Here's my rough attempt to tackle this baby:


Sometimes children from love stem.
You -- now a desert under the sun.
Remember the song that time and again
In the twilight I sung?

Here is a violet, there is a rose
Here is a campfire, there is a hose.
And here is time -- softly waning the day
Taking your mommy and daddy away.

A word play with the popular game of cat and mouse. The metaphore for hide and find, a pray and a victim.

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Note added at 18 mins (2007-07-25 19:05:03 GMT)
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a prey and a predator

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Note added at 10 hrs (2007-07-26 05:08:38 GMT)
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Here's a mouse, here's a cat,
Here's a guard, and here's a camp,
Here's the time that go in circles
Killing slowly mom and dad.

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Note added at 12 hrs (2007-07-26 07:30:07 GMT)
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tjhat runs in circles

Some of the answers already given are brilliant. I am just offering the spider/fly pair as an alternative to cat/mouse.

By the way, I am not aware of an existing translation of this poem, but that does not necessarily mean that there isn't one.

See the spider, see the fly
See the mirador, blocking the sky
See, as Time goes trudging on
Mummy and Daddy dead and gone