A link between magnetars and fast radio burst (FRB) sources has finally been established. In this context, one of the open issues is whether/which sources of extra galactic FRBs exhibit X/gamma-ray outbursts and whether it is correlated with radio activity. We aim to constrain possible X/gamma-ray burst activity from one of the nearest extragalactic FRB sources currently known over a broad energy range, by looking for bursts over a range of timescales and energies that are compatible with being powerful flares from extragalactic magnetars. We followed up the as-yet nearest extragalactic FRB source at a mere 149 Mpc distance, the periodic repeater FRB180916.J0158+65, during the active phase on February 4-7, 2020, with the Insight-Hard X-ray Modulation Telescope (HXMT). Taking advantage of the combination of broad band, large effective area, and several independent detectors available, we searched for bursts over a set of timescales from 1 ms to 1.024 s with a sensitive algorithm, that had previously been characterised and optimised. Moreover, through simulations we studied the sensitivity of our technique in the released energy-duration phase space for a set of synthetic flares and assuming different energy spectra. We constrain the possible occurrence of flares in the 1-100 keV energy band to E<10^46 erg for durations <0.1 s over several tens of ks exposure. We can rule out the occurrence of giant flares similar to the ones that were observed in the few cases of Galactic magnetars. The absence of reported radio activity during our observations does not allow us to make any statements on the possible simultaneous high-energy emission.
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