I don not think there would be a functional relationship between MI and CC since if we consider the formulas for discrete variables for example, we have:

MI=\Sigma_{i}\Sigma_{j} P(x_i,y[_j) log2[P(x_i,y_j)/P(x_i)P(y_j)]  

where P(x,y) is the joint probability mass function and P(x) and P(y) are the marginal probabilities

&

CC=(x*y)_d=\Sigma_{i} (x_i)*y_{i+d}   (non-normalized)

MI is logarithmic while CC is linear.

The extreme cases that you mentioned result from the fact that when the two variables are completely uncorrelated (CC=0), they are not mutually dependent whatsoever (MI=0) and when they are completely correlated (CC=1), they share complete mutual information (MI=1). 

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the two measures have very different properties : mutual information is 0 if and only if the variables are independent ; this is not the case for correlation (independent variables have zero correlation but zero correlation does not imply independence)

such a difference in behaviour leaves little room for a functional dependence

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now, in some restricted fields, you may find such relations ; see for instance

http://www.santafe.edu/media/workingpapers/90-024.pdf

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Fabrice,

You are right: zero correlation does not imply independence but DOES imply zero linear independence. If we assume case in which we wither can have linear dependence or nothing else, then there MUST be a functional dependence of mutual info and correlation.  Would you agree?

Anyway GREAT reference, thanks !

I am not sure i understand

If X and Y are linearly dépendent then corrélation and mutual information are both 1

I guess i am missing something !

Yes, you are wrong :) If two vectors are linearly correlated then their cross corelaton may be between -1 and 1 depending on how strongly they are correlated. That is why we define cross corellation with a lag, so that we can measure the strength of correlation (for that particular lag).