Tendon drive is used for transmission design in applications ranging from wearable rehabilitation and surgical robots to industrial robots. For their efficient operation, one of the principal design requirements is to have adequate tension in the tendon element. This paper introduces a novel mechanism for tension adjustment using a secondary cable-pulley arrangement via a lever mechanism. An adjustment in a tension spring applies a force in the tendon element meant for the transmission of remote centre-of-motion (RCM) mechanism for medical robotic application. Thereafter, the current work presents the kinematic and force analyses of the proposed concept along with a series of plots of various mechanism parameters. The variation of these parameters is in good agreement with the physical motion of the devised mechanism. As compared to the conventional methods of tension adjustment, the proposed concept is a better solution for problems that arise due to low manufacturing tolerances and the need for independent tension adjustment in a multi degrees-of-freedom tendon driven RCM mechanism. In the preferred embodiment, it is proposed to use cables (wire-ropes) as tendon elements to drive a mechanically constrained double-parallelogram-based RCM mechanism which is extensively used for Minimally Invasive Surgery. From this implementation, it is learnt that the presented mechanism can be implemented in several configurations. However, this paper presents the preferred embodiment only which meets the demand, of invariant tendon length, of the RCM mechanism under consideration. An initial prototype is made from acrylic sheets, PLA material, and standard elements (like bearings, pulleys, aluminum pins, 0.5 mm stainless-steel wire-rope, etc.) using laser micromachining and 3D printing.