The US company's attempt to get a judicial review is part of a growing confrontation with New Delhi.
Twitter has been asked by Indian authorities over the past year to act on content including accounts supportive of an independent Sikh state, posts alleged to have spread misinformation about protests by farmers and over tweets critical of the government's handling of the COVID-19 pandemic.
India's IT ministry did not immediately respond on Tuesday to a request for comment about Twitter's legal move.
The Indian government has previously said that big social media firms, including Twitter, have not complied with removal requests, despite their legal standing.
Late last month, Twitter was warned by India's IT ministry of criminal proceedings if it did not comply with some orders. Twitter complied this week, the source said, so as not to lose liability exemptions available as a host of content.
Twitter argues in its request for a judicial review that some removal orders fell short of the procedural requirements of India's IT act, the source said, without specifying which ones Twitter wanted to be reviewed.
The IT act allows the government to block public access to content in the interest of national security, among other reasons.
Twitter, which market research firms say has nearly 24 million users in India, also argues in its filing that some of the orders failed to give notice to authors of the content.
It also says that some were related to political content posted by official handles of political parties, the blocking of which amount to violation of freedom of speech, the source added.
Tensions with the Indian government flared early last year when Twitter declined to fully comply with an order to take down accounts and posts which New Delhi alleged were spreading misinformation about anti-government protests by farmers.
The company has also been subject to police investigations in India and last year many Indian government ministers moved to domestically developed platform Koo, accusing Twitter of non-compliance with local laws.
Twitter has also faced a backlash in India for blocking accounts of influential individuals, including politicians, citing violations of its policies.
India, which industry transparency reports show has among the highest government requests for content takedowns, is considering some amendments to its new IT rules, including the introduction of a government-run appeals panel with the power to reverse the content moderation decisions of social media firms.
New Delhi has said such measures were needed because the companies had violated Indians' constitutional rights.