For two normed algebras $A$ and $B$ with the character space $\bigtriangleup(B)\neq \emptyset$ and a left $B-$module $X,$ a certain class of bounded linear maps from $A$ into $X$ is introduced. We set $CMH_B(A, X)$ as the set of all non-zero $B-$character module homomorphisms from $A$ into $X$. In the case where $\bigtriangleup(B)=\lbrace \varphi\rbrace$ then $CMH_B(A, X)\bigcup \lbrace 0\rbrace$ is a closed subspace of $L(A, X)$ of all bounded linear operators from $A$ into $X$. We define an equivalence relation on $CMH_B(A, X)$ and use it to show that $CMH_B(A, X)\bigcup\lbrace 0\rbrace $ is a union of closed subspaces of $L(A, X)$. Also some basic results and some hereditary properties are presented. Finally some relations between $\varphi-$amenable Banach algebras and character module homomorphisms are examined.