Let $f$ and $g$ be two permutable transcendental entire functions. We shall prove that they have the same Julia set (i.e., $J(f) = J(g)$) if the set of the asymptotic values and critical values of $f$ and $g$ is bounded. This relates to a result and an open problem of Baker in the Fatou-Julia theory. In addition, for any positive integers $n$ and $m$, we show that $J(f \circ g) = J(f^n\circ g^m)$.