  We explore an analogue of the Andr\'e-Oort conjecture for subvarieties of
Drinfeld modular varieties. The conjecture states that a subvariety $X$ of a
Drinfeld modular variety contains a Zariski-dense set of complex multiplication
(CM) points if and only if $X$ is a "special" subvariety (i.e. $X$ is defined
by requiring additional endomorphisms). We prove this conjecture in two cases.
Firstly when $X$ contains a Zariski-dense set of CM points with a certain
behaviour above a fixed prime (which is the case if these CM points lie in one
Hecke orbit), and secondly when $X$ is a curve containing infinitely many CM
points without any additional assumptions.
