The problem of the origin of Near-Earth Objects (NEOs), whether predominantly cometary or asteroidal, has attracted much recent interest. In this work, we consider NEOs to have perihelion distances q<1.4 AU and separate them according to their aphelion distances into two broad classes: Class 1 NEOs have aphelia Q>4.2 AU, allowing these objects possibly to have close approaches to Jupiter; whereas Class 2 objects have Q<4.2 AU. The latter comprise the majority (about 90%) of the known NEO population and are dynamically `decoupled' from close Jovian encounters. This paper presents preliminary results from long-term numerical integrations of hypothetical Jupiter-family comets, which according to these definitions would, if inactive, be considered as Class 1 `cometary' NEOs, or `cometary asteroids'. In particular, we evaluate the transfer probabilities to Class 2 orbits, by both gravitational and non-gravitational mechanisms, and estimate the overall cometary contribution to the number of objects in Class 1 and Class 2. Our results are compared with estimates of the rate of injection of NEOs from the main asteroid belt.