Necessary and sufficient conditions in terms of operator polynomials are obtained for an
m
m
-tuple
T
=
(
T
1
,
…
,
T
m
)
T = ({T_1}, \ldots ,{T_m})
of commuting bounded linear operators on a separable Hilbert space
H
\mathcal {H}
to extend to an
m
˙
\dot m
-tuple
S
=
(
S
1
,
…
,
S
m
)
S = ({S_1}, \ldots ,{S_m})
of operators on some Hilbert space
K
\mathcal {K}
, where each
S
i
{S_i}
is realized as a
∗
{\ast }
-representation of the adjoint of a multiplication operator on the tensor product of a special type of functional Hilbert spaces. Also, necessary and sufficient conditions in terms of operator polynomials are obtained for
T
T
to have a commuting normal extension.