This paper presents an approximate wireless communication scheme for federated learning (FL) model aggregation in the uplink transmission. We consider a realistic channel that reveals bit errors during FL model exchange in wireless networks. Our study demonstrates that random bit errors during model transmission can significantly affect FL performance. To overcome this challenge, we propose an approximate communication scheme based on the mathematical and statistical proof that machine learning (ML) model gradients are bounded under certain constraints. This bound enables us to introduce a novel encoding scheme for float-to-binary representation of gradient values and their QAM constellation mapping. Besides, since FL gradients are error-resilient, the proposed scheme simply delivers gradients with errors when the channel quality is satisfactory, eliminating extensive error-correcting codes and/or retransmission. The direct benefits include less overhead and lower latency. The proposed scheme is well-suited for resource-constrained devices in wireless networks. Through simulations, we show that the proposed scheme is effective in reducing the impact of bit errors on FL performance and saves at least half the time than transmission with error correction and retransmission to achieve the same learning performance. In addition, we investigated the effectiveness of bit protection mechanisms in high-order modulation when gray coding is employed and found that this approach considerably enhances learning performance.