In the work, we develop a least-squares finite element method for linear Phan-Thien–Tanner (PTT) viscoelastic fluid flows; in contrast to the Newtonian flows, the problems are associated with fluid viscosity and elasticity. We consider the least-squares finite element method with stabilized weights for the viscoelastic model and prove that the LS approximation converges to the linearized solutions of the linear PTT model. An a posteriori error estimator of the LS functional is used for an adaptive weight iteration approach. This approach improves mass conservation and yields convergence at high Weissenberg numbers when low order basis functions. For numerical experiments, we first consider the flow through a planar channel to illustrate our theoretical results. The LS method is then applied to a flow through the slot channel with two depth ratios and the effects of physical parameters are discussed. Numerical solutions of the channel problem indicate that flow characteristics of the viscoelastic polymer solution are described by the results obtained using the method. Furthermore, we present the hole pressure for various Weissenberg numbers, and compare with that derived from the Higashitani–Pritchard (HP) theory.