Despite the impressive performance of large language models across various tasks, they often struggle with reasoning under negated statements. Negations are important in real-world applications as they encode negative polarity in verb phrases, clauses, or other expressions. Nevertheless, they are underrepresented in current benchmarks, which mainly include basic negation forms and overlook more complex ones, resulting in insufficient data for training a language model. In this work, we propose NegVerse, a method that tackles the lack of negation datasets by producing a diverse range of negation types from affirmative sentences, including verbal, non-verbal, and affixal forms commonly found in English text. We provide new rules for masking parts of sentences where negations are most likely to occur, based on syntactic structure and use a frozen baseline LLM and prompt tuning to generate negated sentences. We also propose a filtering mechanism to identify negation cues and remove degenerate examples, producing a diverse range of meaningful perturbations. Our results show that NegVerse outperforms existing methods and generates negations with higher lexical similarity to the original sentences, better syntactic preservation and negation diversity. The code is available in https://github.com/DarianRodriguez/NegVerse