Multilingual Large Language Models (LLMs) struggle with cross-lingual tasks due to data imbalances between high-resource and low-resource languages, as well as monolingual bias in pre-training. Existing methods, such as bilingual fine-tuning and contrastive alignment, can improve cross-lingual performance, but they often require extensive parallel data or suffer from instability. To address these challenges, we introduce a Cross-Lingual Mapping Task during the pre-training phase, which enhances cross-lingual alignment without compromising monolingual fluency. Our approach bi-directionally maps languages within the LLM embedding space, improving both language generation and comprehension. We further propose a Language Alignment Coefficient to robustly quantify cross-lingual consistency, even in limited-data scenarios. Experimental results on machine translation (MT), cross-lingual natural language understanding (CLNLU), and cross-lingual question answering (CLQA) show that our model achieves gains of up to 11.9 BLEU points in MT, 6.72 points in CLQA BERTScore-Precision, and more than 5% in CLNLU accuracy over strong multilingual baselines. These findings highlight the potential of incorporating cross-lingual objectives into pre-training to improve multilingual LLMs.