Let U v ( sl n ) be the quantized enveloping algebra of sl n over C ( v ) and Ω v be the natural representation of U v ( sl n ) , where v is an indeterminate. There is a natural right action of the Hecke algebra H v ( S r ) of the symmetric group S r on Ω v ⊗ r , commuting with the action of U v ( sl n ) . It is well known that the natural algebra homomorphisms ζ r : U v ( sl n ) → S v ( n , r ) and η r : H v ( S r ) op → End U v ( sl n ) ( Ω v ⊗ r ) are surjective, where S v ( n , r ) = End H v ( S r ) ( Ω v ⊗ r ) is the quantum Schur algebras over C ( v ) . Given an associative algebra A , let Z ( A ) be the center of A . In this paper, we prove that the maps ζ r and η r preserve the center. That is, we prove that ζ r ( Z ( U v ( sl n ) ) ) = Z ( Im ζ r ) = Z ( S v ( n , r ) ) and η r ( Z ( H v ( S r ) op ) ) = Z ( Im η r ) . It should be noted that when v is specialized to a primitive root of unity ε in C , ζ r , ε ( Z ( U ε ( sl n ) ) ) is not equal to Z ( Im ζ r , ε ) in general, where U ε ( sl n ) is quantum sl n over C at parameter ε . Finally, we use bases of the center of the Hecke algebra H v ( S r ) to construct bases of the center of the quantum Schur algebra S v ( n , r ) .