We explore the prospects for a population inversion in the hyperfine levels of atomic hydrogen at high redshifts when the Universe was still predominanly neutral. The existence of 21-cm masers during the epoch of reionisation (EoR) could greatly boost the expected EoR signals. We show that the color temperature of Lya photons emitted by a bright source embedded in a neutral collapsing gas cloud may be sufficiently negative to establish population inversion in the hyperfine transition via the Wouthuysen-Field effect. In practice, however, a small flux of Lya photons produced via x-ray heating or through cascades of atoms which are photo-excited by higher Lyman-series photons, can counteract this effect. In order for maser conditions to prevail, the spectrum of the source must be strongly suppressed blueward of the Ly-beta transition. We specify conditions in a clumpy two-phase interstellar medium (ISM), which could give rise to the required spectrum. We find that the most likely sites for 21-cm masers are cold infall regions around galaxies undergoing massive star formation (> 10^3 Msun/yr), obscured from view by a large column of gas with NH > 10^22.5/cm^2 (reminiscent of known starburst galaxies at lower redshifts). Crucial for the existence of masers is for the dust in the ISM to be locked-up in cold dense clumps embedded in a hot tenous medium, through which a large fraction of the emitted Lya photons can escape the galaxies.