The bill was tabled ahead of a national election that is expected to be called within weeks and as Najib faces widespread criticism over the scandal at state fund 1Malaysia Development Berhad (1MDB).
Under the Anti-Fake News 2018 bill, anyone who published so-called fake news could face fines of up to 500,000 ringgit ($128,140), up to 10 years in jail, or both.
"The proposed Act seeks to safeguard the public against the proliferation of fake news whilst ensuring the right to freedom of speech and expression under the Federal Constitution is respected," it said.
It defines fake news as "news, information, data or reports which is or are wholly or partly false" and includes features, visuals and audio recordings.
The law, which covers digital publications and social media, also applies to offenders outside Malaysia, including foreigners, as long as Malaysia or a Malaysian citizen were affected.