We have used the Australia Telescope Compact Array (ATCA) to search for a number of centimetre wavelength methanol transitions which are predicted to show weak maser emission towards star formation regions. Sensitive, high spatial and spectral resolution observations towards four high-mass star formation regions which show emission in a large number of class II methanol maser transitions did not result in any detections. From these observations we are able to place an upper limit of <~1300K on the brightness temperature of any emission from the $3_1$A$^+$-$3_1$A$^-$, $17_{-2}$-$18_{-3}$E ($v_t=1$), $12_4$-$13_3$A$^-$, $12_4$-$13_3$A$^+$ and $4_1$A$^+$-$4_1$A$^-$ transitions of methanol in these sources on angular scales of 2 arcseconds. This upper limit is consistent with current models for class II methanol masers in high-mass star formation regions and better constraints than those provided here will likely require observations with next-generation radio telescopes.
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