Studies using the Energetic Gamma-Ray Experiment Telescope (EGRET) revealed that blazars [flat-spectrum radio quasars (FSRQs) and BL Lac objects] emit most of their luminosity in the high-energy gamma-ray (E > 100 MeV) range. From the 271 sources observed by EGRET, 131 are still unidentified. A systematic search is conducted to identify possible high-energy gamma-ray blazars among the unidentified EGRET population. Based upon multiwavelength emission properties, 13 extragalactic radio sources were selected in the EGRET error boxes for further investigation. From the above-mentioned sample, results of a multiwavelength follow-up of two EGRET sources, 3EG J0821−5814 and 3EG J0706−3837, are presented. These sources are associated with their radio counterparts PKS J0820−5705 and PMN J0710−3850, respectively. Spectroscopic observations utilizing the SOAR/Goodman spectrograph at the Cerro Tololo Inter-American Observatory in Chile reveal a spectrum of PKS J0820−5705 that corresponds to that of a radio-loud active galactic nucleus (FSRQ) with redshift z = 0.06 ± 0.01, while the visibility of wide and narrow emission lines in the spectrum of PMN J0710−3850 resembles that of a low-ionization nuclear emission-line region (LINER) or type 1 Seyfert galaxy at z = 0.129 ± 0.001. The observed Ca ii K&H lines depression ratio at 4000 Å showed a shallow depression of 8.8 ± 2.5 per cent for PKS J0820−5705 and 80 ± 1 per cent for PMN J0710−3850, suggesting the presence of a strong non-thermal optical contribution in PKS J0820−5705, which clearly distinguishes its spectrum from that of a radio galaxy. The weaker optical non-thermal contribution for PMN J0710−3850 is in accordance with that expected of a LINER. For PMN J0710−3850 the line flux ratios [O iii] λ5007/Hβ < 3 and [N ii] λ6583/Hα > 0.6 which are in agreement with the expected ratios of LINERs. However, the absence of [O ii] λ3727 implies an anomalously low [O ii]/[O iii] < 0.5 ratio for a LINER, and agrees more with the ratio observed in type 1 Seyfert galaxies. The average velocities inferred from the Balmer lines range between 2300 and 4300 km s−1, while [O i] and [O iii] velocities range between 420 and 490 km s−1, consistent with both LINERs and type 1 Seyfert galaxies. The X-ray luminosities of these two sources are LX ∼ 9 × 1043 erg s−1 (PKS J0820−5705) and LX ∼ 9 × 1042 erg s−1 (PMN J0710−3850), respectively. The X-ray luminosity of PMN J0710−3850 is an order of magnitude higher than the upper limit detected from LINERs, and correlates well with the typical X-ray luminosities observed in type 1 Seyfert galaxies. The X-ray luminosity of PKS J0820−5705 is consistent with the observed luminosity of FSRQs. Optical photometry carried out with the South African Astronomical Observatory 1.0-m telescope displayed 1–2 mag variability in the B and R bands for PKS J0821−5705, on time-scales of hours, while a 5σ variability of the average R-band magnitude could be discerned over a 3 d time span. A smaller 0.5 mag variability is visible in the B band for PMN J0710−3850 on time-scales of hours. No variability was detected in the R band for this source.