As part of the long-term radial velocity monitoring of known transiting planets, we have acquired new radial velocity data for the two transiting systems WASP-12 and WASP-14, each harbouring a gas giant on a close orbit (orbital period of 1.09 and 2.24 d, respectively). In both cases, the initial orbital solution suggested a significant orbital eccentricity, 0.049 ± 0.015 for WASP-12b and 0.091 ± 0.003 for WASP-14b. Since then, measurements of the occultation of WASP-12 in the infrared have indicated that one projection of the eccentricity (e cos ω) was close to zero, casting doubt on the eccentricity from the initial radial velocity orbit. Our measurements show that the radial velocity data are compatible with a circular orbit. A MCMC analysis taking into account the presence of correlated systematic noise in both the radial velocity and photometric data gives e= 0.017+0.015−0.010. In contrast, we confirm the orbital eccentricity of WASP-14b, and refine its value to e= 0.0877 ± 0.0030, a 10σ detection. WASP-14b is thus the closest presently known planet with a confirmed eccentric orbit.