What is the correct expression: "the estimate number" or "the estimated number?"

Solution 1:

The Corpus of Contemporary American English attests 124 instances of estimated number and but a single instance of estimate number.

This solitary hit is a false positive, because here estimate is a verb, not an attributive noun. The citation is from the New York Times in 2012:

. . . clear Marja, a district of 100 square miles, of Taliban insurgents that residents estimate number no more than 200.

So there is no recorded use of estimate as an attributive noun used before number in COCA at all.

Solution 2:

To me, the two mean different things.

An estimated number is any number that has been estimated, guessed at, roughly calculated, etc.

The estimated number of casualties in the plane crash exceeded 200.

An estimate number, on the other hand, is a much narrower and less applicable notion: it is specifically a number mentioned in an estimate (as defined in sense 1.2 of the ODO entry):

A written statement indicating the likely price that will be charged for specified work or repairs.

If you receive an estimate for the delivery of 2,000 items, but only 1,500 are shipped to you for the full price given in the estimate, you might want to lodge a complaint with the person or company who gave you the estimate. In that complaint, you may write something like:

The actual number of items received is only 1,500, though the estimate number is 2,000.

(Quite possibly, you do not want to write anything like that, because it’s fairly clumsy. “… although the estimate clearly says 2,000 items were to be shipped” is much less cumbersome, if a bit longer.)

Since this latter sense is so very limited, and since I cannot think of a way to work an extra of onto the end of it, I can only imagine that what you’re after is an estimated number (of somethings).