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We study the emergence of precessing vortex core (PVC) oscillations in a swirling jet experiment. We vary the swirl intensity while keeping the net mass flow rate fixed using a radial-entry swirler with movable blades upstream of the jet exit. The swirl intensity is quantified in terms of a swirl number $$S$$ . Time-resolved velocity measurements in a radial–axial plane anchored at the jet exit for various $$S$$ values are obtained using stereoscopic particle image velocimetry. Spectral proper orthogonal decomposition and spatial cross-spectral analysis reveal the simultaneous emergence of a bubble-type vortex breakdown and a strong helical limit-cycle oscillation in the flow for $$S>S_{c}$$ where $$S_{c}=0.61$$ . The oscillation frequency, $$f_{PVC}$$ , and the square of the flow oscillation amplitudes vary linearly with $$S-S_{c}$$ . A solution for the coherent unsteady field accurate up to $$O(\unicode[STIX]{x1D716}^{3})$$ ( $$\unicode[STIX]{x1D716}\sim O((S-S_{c})^{1/2})$$ ) is determined from the nonlinear Navier–Stokes equations, using the method of multiple scales. We show that onset of bubble type vortex breakdown at $$S_{c}$$ , results in a marginally stable, helical linear global hydrodynamic mode. This results in the stable limit-cycle precession of the breakdown bubble. The variation of $$f_{LC}$$ with $$S-S_{c}$$ is determined from the Stuart–Landau equation associated with the PVC. Reasonable agreement with the corresponding experimental result is observed, despite the highly turbulent nature of the flow in the present experiment. Further, amplitude saturation results from the time-averaged distortion imposed on the flow by the PVC, suggesting that linear stability analysis may predict PVC characteristics for $$S>S_{c}$$ .