Fluid-structure interaction (FSI) phenomena play a pivotal role in diverse engineering applications such as aerodynamics, civil engineering, mechanical engineering, and biomechanics. This involves intricate physical interactions where fluids engage with elastic structures, including scenarios like blood flow in arteries, fluid dynamics in turbines, and structural responses to wind loads in aircraft wings, towers, and bridges. This study presents a novel approach to monolithically coupled fluid-structure interaction utilizing mixed least-squares (LS) stress-velocity (SV) formulations with implicit time discretization schemes. The mixed least-squares finite element method (LSFEM) employed here surpasses limitations associated with the Ladyzhenskaya–Babuska–Brezzi (LBB) condition and exhibits theoretical advantages over the established (mixed) Galerkin method, as discussed in [1]. Specifically, the LSFEM yields positive definite and symmetric system matrices, even for differential equations with non-self-adjoint operators, see e.g. [2]. The variational approach for the fluid involves the incompressible Navier-Stokes equations in an Arbitrary-Lagrangian-Eulerian (ALE) description to account for a deformable fluid domain, as explained in [3]. For the deformation of the fluid background mesh, a pseudo-material with linear elastic behavior and local hardening is used. The proposed LS solid formulation express stresses and velocities with conforming finite element spaces in H(div) and H1 (same as the fluid) for spatial discretization of the unknowns, ensuring inherent fulfillment of the coupling conditions in FSI problems. To validate the effectiveness of the coupled LS FSI approach, we conduct tests by solving a benchmark problem involving flow around a cylinder with an attached flag. This scenario is characterized by substantial deformations of the solid flag and, consequently, the fluid domain. [2] P.B. Bochev and M.D. Gunzburger. Least-Squares Finite Element Methods. Springer-Verlag, New York, 1st edition, 2009. [2] Z. Cai, B. Lee, and P. Wang. Least-squares methods for incompressible Newtonian fluid flow: Linear stationary problems, \textit{SIAM J. Numer. Anal.}, 42(2):843-859, 2004. [3] J. Donea, A. Huerta, J.-P. Ponthot, and A. Rodriguez-Ferran. Arbitrary Lagrangian- Eulerian Methods, chapter 14. John Wiley and Sons, Ltd, 2004.