A tensioning system of a mooring line of a floating structure. The mooring line includes separate first and second mooring line sections, with a first end of the second mooring line attached to the floating structure and a first end of the first mooring line section being attached to an anchoring device embedded in a seabed. A tensioner configured to act on the mooring line includes a body with a housing, and an inlet unit for inserting a second end of the first mooring line section into the housing of the body. The body includes an opening for the passage of a second end of the first mooring line section from the inlet unit, through the housing, and along a direction changing element. The tensioning system includes a stabilizing platform fixed to the body. At least a portion of the stabilizing platform laterally projects from the body of the tensioner.