The hydrodynamic force resisting the relative motion of two drops moving along their line‐of‐centers is determined for Stokes flow conditions. The drops are assumed to be in near‐contact and to have sufficiently high interfacial tension that they remain spherical. The squeeze flow in the narrow gap between the drops is analyzed using lubrication theory, and the flow within the drops near the axis of symmetry is analyzed using a boundary integral technique. The two flows are coupled through the nonzero tangential stress and velocity at the interface.Depending on the ratio of drop viscosity to that of the continuous phase, and also on the ratio of the distance between the drops to their reduced radius, three possible flow situations arise, corresponding to nearly rigid drops, drops with partially mobile interfaces, and drops with fully mobile interfaces. The results for the resistance functions are in good agreement with an earlier series solution using bispherical coordinates. The new results for near‐contact motion have important implications for droplet collisions and coalescence.The theory is also extended to consider the normal motion of a drop toward a solid boundary. As expected, the resistance to this motion is less than that experienced by a solid particle moving toward a solid boundary. In particular, the force on a drop as it becomes very close to the boundary approaches one‐fourth of that on a solid sphere with the same size and relative velocity.