The widespread use of large language models (LLMs) influences different social media and educational contexts through the overwhelming generated text with a certain degree of coherence. To mitigate their potential misuse, this paper explores the feasibility of finetuning LLaMA with label supervision (named LS-LLaMA) in unidirectional and bidirectional settings, to discriminate the texts generated by machines and humans in monolingual and multilingual corpora. Our findings show that unidirectional LS-LLaMA outperformed the sequence language models as the benchmark by a large margin. Our code is publicly available at https://github.com/honghanhh/llama-as-a-judge.