A bounded linear operator $T$ is called  $\infty$-hypo\-normal 
if $T$ is $p$-hyponormal for every $p>0$. 
In this paper $\infty$-hyponormality of the Aluthge transformations  
of $\infty$-hyponormal operators is investigated. 
It is shown that 
the Aluthge transformation of an $\infty$-hyponormal operator is not 
necessarily $\infty$-hyponormal. 
It is also shown that the  (generalized)  Aluthge transformation 
 of an $\infty$-hyponormal operator $T$  
is  $\infty$-hyponormal provided   $|T||T^\ast|=|T^\ast||T|$. 
Moreover we give an example of an $\infty$-hyponormal operator $T$  
whose Aluthge transformation $\tilde{T}$ is  $\infty$-hyponormal 
but $|T||T^\ast|\not=|T^\ast||T|$.