We in this note consider a new concept, so called $pi$-McCoy, which unifies McCoy rings and IFP rings. The classes of McCoy rings and IFP rings do not contain full matrix rings and upper (lower) triangular matrix rings, but the class of $pi$-McCoy rings contain upper (lower) triangular matrix rings and many kinds of full matrix rings. We first study the basic structure of $pi$-McCoy rings, observing the relations among $pi$-McCoy rings, Abelian rings, 2-primal rings, directly finite rings, and ($pi-$)regular rings. It is proved that the n by n full matrix rings ($ngeq2$) over reduced rings are not $pi$-McCoy, finding $pi$-McCoy matrix rings over non-reduced rings. It is shown that the $pi$-McCoyness is preserved by polynomial rings (when they are of bounded index of nilpotency) and classical quotient rings. Several kinds of extensions of $pi$-McCoy rings are also examined.