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This study extended the conventional JKR theory on adhesive contact between elastic cylinders to the frictionless adhesive contact between elastic films coated on rigid cylindrical rollers. The plane-strain elasticity problem of indentation on an elastic film perfectly bonded to a rigid half-plane was revisited, with which the force-depth relations for both flat and cylindrical indentations were obtained in a simple form. By following Hertz analysis and using the solution of the cylindrical indentation on an elastic film, the adhesionless Hertzian contact between elastic films coated on rigid cylinders was obtained. However, the obtained Hertzian contact state should be considered as an approximate solution, since the pressure distributions in the contact region of the two elastic films may not match perfectly each other in the case of different Poisson’s ratios or thicknesses of the elastic films. By properly superposing the linear elastic solutions of flat and cylindrical indentations, the approximate solution to the adhesive JKR contact state between elastic films coated on rigid cylinders was also obtained, of which the relations between applied force, penetration depth, and contact width were expressed in terms of the geometric parameters and elastic constants of the two elastic films. The pull-off force at the moment of debonding two elastic films was also obtained as a function of the adhesion energy and compared with the normal two-dimensional JKR result.