We propose a sub-band spreading technique for adaptive modulation (AM) in orthogonal frequency division multiplexing (OFDM) systems in order to reduce signaling overheads and to average frequency selective fading channels causing different signal-to-noise ratio (SNR) values for subcarriers in each subband. The conventional sub-band based AM schemes can also reduce signaling overheads and complexity for allocating a resource per sub-band at a time. However, they may suffer from the channel variation in a sub-band when the sub-band size is larger than the channel coherence bandwidth (BW). The sub-band spreading at the transmitter enables the received symbols in each sub-band to have an identical reliability even in a frequency selective fading channel. We rigorously analyze the averaged SNR value at the receiver of the sub-band spreading system and the analyzed average SNR in a sub-band is used for an adaptation criterion. The proposed AM scheme outperforms the conventional sub-band based OFDM scheme without spreading, and it can yield better throughput performance than the conventional subcarrier based AM schemes when we consider the signaling overheads.