Large language models (LLMs) have achieved state-of-the-art performance in various language processing tasks, motivating their adoption in simultaneous translation. Current fine-tuning methods to adapt LLMs for simultaneous translation focus on prompting optimization strategies using either data augmentation or prompt structure modifications. However, these methods suffer from several issues, such as an unnecessarily expanded training set, computational inefficiency from dumping the KV cache, increased prompt sizes, or restriction to a single decision policy. To eliminate these issues, we propose a new paradigm in fine-tuning LLMs for simultaneous translation, called SimulMask. It utilizes a novel attention mask technique that models simultaneous translation during fine-tuning by masking attention connections under a desired decision policy. Applying the proposed SimulMask on a Falcon LLM for the IWSLT 2017 dataset, we have observed a significant translation quality improvement compared to state-of-the-art prompting optimization strategies on three language pairs when averaged across four different latency regimes while reducing the computational cost.