We introduce the operators which are generalizations of Hankel-type operators, called the $(C, r)$-Hankel operator and $(R, r)$-Hankel operator on general Hilbert spaces. Our main result is to obtain characterizations for a bounded operator on general Hilbert spaces to be a $(C, r)$-Hankel operator (or $(R, r)$-Hankel operator). We also discuss some algebraic properties like boundedness (for $|r| \neq 1$) of these operators and the relationship between them. Moreover, some characterizations for the commutativity of these operators are explored.