  In a step reinforced random walk, at each integer time and with a fixed
probability p $\in$ (0, 1), the walker repeats one of his previous steps chosen
uniformly at random, and with complementary probability 1 -- p, the walker
makes an independent new step with a given distribution. Examples in the
literature include the so-called elephant random walk and the shark random
swim. We consider here a continuous time analog, when the random walk is
replaced by a L{\'e}vy process. For sub-critical (or admissible) memory
parameters p < p c , where p c is related to the Blumenthal-Getoor index of the
L{\'e}vy process, we construct a noise reinforced L{\'e}vy process. Our main
result shows that the step-reinforced random walks corresponding to discrete
time skeletons of the L{\'e}vy process, converge weakly to the noise reinforced
L{\'e}vy process as the time-mesh goes to 0.
