Oscillatory behaviour in the tryptophan operon of an Escherichia coli mutant strain lacking the enzyme-inhibition regulatory mechanism has been observed by Bliss et al. but not confirmed by others. This behaviour could be important from the standpoint of synthetic biology, whose goals include the engineering of intracellular genetic oscillators. This work is devoted to investigating, from a mathematical modelling point of view, the possibility that the trp operon of the E. coli inhibition-free strain expresses cyclically. For that we extend a previously introduced model for the regulatory pathway of the tryptophan operon in Escherichia coli to account for the observed multiplicity and cooperativity of repressor binding sites. Thereafter we investigate the model dynamics using deterministic numeric solutions, stochastic simulations, and analytic studies. Our results suggest that a quasi-periodic behaviour could be observed in the trp operon expression level of single bacteria.