This paper deals with near-optimal spectrum sensing of orthogonal frequency division multiplexing (OFDM) signals to achieve reliable detection at low signal-to-noise ratio (SNR). The likelihood ratio test (LRT), a simple hypothesis test, delivers the optimal performance, but it requires that the parameters involved in the test are known. Hence, the generalized likelihood ratio test (GLRT), a composite hypothesis test, has often been employed. However, GLRT-based detectors involve biased parameter estimation, which may lead to inferior performance. In this paper, with proper approximation, the LRT is reduced to a simpler form, which only requires the knowledge of noise power. Then a novel unbiased and consistent estimator of the noise power is developed. This estimator is combined with the approximate LRT, leading to an asymptotic simple hypothesis test (ASHT). Two ASHT-based detectors are presented for cases with and without time synchronization, and their theoretical performances are analyzed. Simulation results show that the ASHT-based detectors deliver performances very close to that of the optimal LRT, and significantly outperform the existing GLRT-based detectors. It is also shown that the ASHT-based detectors exhibit robustness against the influence of multipath channels.