Two blind adaptive receivers, based on constrained constant modulus (CM) algorithms, are investigated for the detection of the multicarrier code division multiple access (MC-CDMA) system in the multiuser and multipath channels. As the data rate of the information bit is getting high enough, the transmitted signal of the MC-CDMA system becomes more sensitive to the frequency-selective fading. To mitigate this drawback, the parallel transmissions of multiple data symbols within one OFDM block are employed in the proposed systems to ensure that the length of the cyclic prefix (CP) is longer than the delay spread of the multipath channel. The first receiver is called a full-tap receiver (TDes receiver), which can carry out the interference suppression in the frequency-domain. Alternatively, a reduced-tap receiver (TRTap) is proposed on the basis of the derived cyclically shifting matrix. According to the derived matrix representation of the received signal, the analytical steady-state mean square errors (MSEs) for both receivers are also derived and compared. Simulation results are provided to illustrate the effectiveness of the proposed receivers and validate the accuracy of the theoretical analysis. As expected, the symbol error rate (SER) performance of the full-tap receiver is better than that of the reduced-tap receiver in severe interference environments. In contrast, the reduced-tap receiver can provide the satisfied SER performance just like the full-tap receiver in low-interference scenarios, even though the former one adopts only one-tap equalization.