The discovery may overturn ideas of when relatively advanced life evolved, say scientists in Sweden.
They identified parts of chloroplasts, structures within plant cells involved in photosynthesis.
The earliest signs of life on Earth are at least 3.5 billion years old.
The first single-celled microscopic life forms evolved into larger multi-cellular eukaryotic organisms (made up of cells containing a nucleus and other structures within a membrane).
Therese Sallstedt of the Swedish Museum of Natural History discovered some of the fossils. She described them as "the oldest fossil plants that we know of on Earth in the form of 1.6 billion year old red algae".
"They show us that advanced life in the form of eukaryotes (like plants, fungi and us humans/animals) have a much deeper history on Earth than what we previously have thought," she told BBC News.
Tree of life
The scientists found thread-like fossils and more complex "fleshy" colonies in sedimentary rock from central India. Both have characteristics of modern red algae, a type of seaweed.
Co-researcher Prof Stefan Bengtson of the Swedish Museum of Natural History added: "You cannot be 100% sure about material this ancient, as there is no DNA remaining, but the characters agree quite well with the morphology and structure of red algae."
The oldest known red algae before the present discovery date back 1.2 billion years. The Indian fossils are 400 million years older, suggesting that the early branches of the tree of life began much earlier than previously thought.
Claims of ancient life are always controversial. Without DNA evidence, confirmation must rest on whether more fossils can be found.
There is also debate over whether red algae belong in the plant kingdom or in a class of their own.
Modern red algae is perhaps best known for two commercial products - gelatinous texturing agents used in making ice cream - and nori - the seaweed used to wrap sushi.
/BBC/
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