ABSTRACT Observations of the redshifted 21-cm hyperfine line of neutral hydrogen from early phases of the Universe such as Cosmic Dawn and the Epoch of Reionization promise to open a new window onto the early formation of stars and galaxies. We present the first upper limits on the power spectrum of redshifted 21-cm brightness temperature fluctuations in the redshift range z = 19.8–25.2 (54–68 MHz frequency range) using 14 h of data obtained with the LOFAR-Low Band Antenna (LBA) array. We also demonstrate the application of a multiple pointing calibration technique to calibrate the LOFAR-LBA dual-pointing observations centred on the North Celestial Pole and the radio galaxy 3C220.3. We observe an unexplained excess of $∼ 30\!-\!50{{\ \rm per\ cent}}$ in Stokes / noise compared to Stokes V for the two observed fields, which decorrelates on ≳12 s and might have a physical origin. We show that enforcing smoothness of gain errors along frequency direction during calibration reduces the additional variance in Stokes I compared Stokes V introduced by the calibration on sub-band level. After subtraction of smooth foregrounds, we achieve a 2σ upper limit on the 21-cm power spectrum of $Δ _{21}^2 \lt (14561\, \text{mK})^2$ at $k∼ 0.038\, h\, \text{cMpc}^{-1}$ and $Δ _{21}^2 \lt (14886\, \text{mK})^2$ at $k∼ 0.038 \, h\, \text{cMpc}^{-1}$ for the 3C220 and NCP fields respectively and both upper limits are consistent with each other. The upper limits for the two fields are still dominated by systematics on most k modes.