Hilbert's lecture at the ICM in Paris in 1900 presented 10 of the famous 23 open problems. It is well-known that the idea of the lecture came from Hermann Minkowski. Hilbert was at Gottingen at the time where he was hired through untiring efforts of Felix Klein. As detailed by historian David Rowe and others, both Hilbert and Klein were involved in a battle against the Berliners at the time. The Berlin school dominated by followers of Kummer, Weierstrass, and Kronecker was known for its focus on arithmetized analysis. Hilbert's 23 open problems sought to broaden the scope of mathematics beyond such narrow focus. It seems as though it would have been natural for Hilbert to have discussed the 23 problems with Klein. Is there any evidence of such discussions in published work or private correspondence?

Here is what Minkowski wrote:

"Most alluring would be the attempt to look into the future, in other words, a characterization of the problems to which the mathematicians should turn in the future. With this, you might conceivably have people talking about your speech even decades from now. Of course, prophecy is indeed a difficult thing" (Minkowski 1973, 5 January 1900). 

The reference is 

Hermann Minkowski, Briefe an David Hilbert, Hg. L. Ru¨denberg und H. Zassenhaus, New York: Springer-Verlag, 1973. 

This information comes from page 16 of Rowe's article 

Rowe, D. "Mathematics made in Germany: on the background to Hilbert's Paris lecture." Math. Intelligencer 35 (2013), no. 3, 9--20.