The paper studies a certain class of rational maps and the goal is to provide necessary and sufficient conditions for the Julia set of such maps to be a Jordan arc. The class under consideration is the conjugate under the Joukowski map \(J(z)=\frac{1}{2} (z+\frac{1}{z})\) of a product \(g_{n,\nu}^\pm=\pm z^{n-\nu} \prod_{k=1}^{\nu}\frac{z-z_k}{1-zz_k}\) for some positive integer \(n\) and it appears denoted as \(\mathcal{R}_n^\pm(y,\rho_\nu)=J\circ g_{n,\nu}^\pm\circ J^{-1}(y)\), where \(\rho_\nu(z)= \prod_{k=1}^{\nu}(z-z_k)\). The results are obtained for a slightly more general class, but in all cases, the dynamical behavior is given by the product \(g_{n,\nu}^\pm(z)\), which is a perturbation of a Blaschke product. The Julia set of a Blaschke product is the unit circle \(\mathbb{D}\). Under a very weak condition that the zeros of \(g_{n,\nu}^\pm\) stay in a disc \(\{|z|\leq r\}\), where \(r\) is dependent on \(\nu\) and \(n\), it follows that \(|g_{n,\nu}^\pm(z)|<|z|\) for \(|z|\leq r\). Thus, all orbits starting in the attracting basin at 0 converge to 0, which means that this basin is completely invariant and so is the basin of \(\infty\), by the \((1/z)\)-symmetry. With two completely invariant Fatou components, the Julia set is indeed a Jordan arc (Theorem 2.3). By contrast, when the zeros are somewhat scattered outside an annulus centered at 0, it may happen that the Julia set of \(g_{n,\nu}^\pm\) is not even connected. An example may be found among a sequence of such \(g_{n,\nu}^\pm\)'s, as proved in Theorem 2.5. A necessary condition for the Julia set to be a Jordan arc is given in Section 3, where it is proved that any rational map of degree \(\geq2\) with Julia set a Jordan arc has the form \(\mathcal(R)_n^\pm(y,\rho_\nu)\). In Section 4, a class of rational functions with a rationally indifferent fixed-point is given for which the Julia set is a Jordan arc. First, a transformed Blaschke product by the map \(M(z)=\frac{1+z}{1-z}\) yields a rational function with only one rationally indifferent fixed point at \(\infty\) (Lemma 4.3). This function is then perturbed by an odd polynomial to give a rational map with a rationally indifferent fixed point at \(\infty\) and with Julia set a Jordan arc (Theorems 4.5 and 4.6). These final results allow one to describe a more general class of rational functions whose Julia set is a Jordan arc, which are similar to \(\mathcal{R}_n^\pm(y,\rho_\nu)\), but which may have rationally indifferent fixed points. They are of the form \[ \mathcal{R}_n^\pm(y,\rho_\nu,a,b)= \frac{a+b}{2}+\frac{a-b}{2} \mathcal{R}_n^\pm\left(\frac{2y}{a-b}-\frac{a+b}{a-b},\rho_\nu\right), \] with \(a,b\in\mathbb{C},a\neq b\).