Graphical methods are a key tool to analyze Lur’e systems with time delay. In this letter we revisit clockwise properties of the Nyquist plot and extend results in the literature to critically stable systems and time-delayed systems. It is known that rational transfer functions with no resonant poles and no zeros satisfy the Kalman conjecture. We show that the same class of transfer functions in series with a time delay also satisfies the Kalman conjecture. Furthermore the same class of transfer functions in series with an integrator and delay (which may be zero) satisfies a suitably relaxed form of the Kalman conjecture. Useful results are also obtained where the delay is constant but unknown. Results in this letter can be used as benchmarks to test sufficient stability conditions for the Lur’e problem with time-delay systems