"Now I would like to focus on large projects that are related to the development of private initiatives in a broad sense, but at the same time have a significant impact on people."
He held the post since 2018, appointed after Putin was re-elected for a fourth term in the Kremlin.
Last week, Russian state news agency TASS reported that Kudrin will be given a senior role in Yandex before the new year.
"The decision has been taken," TASS quoted a source as saying.
Yandex -- often dubbed the "Russian Google" -- has a search engine and also provides Russia's main taxi and food delivery service.
It is registered in the Netherlands and has European, UK and US subsidiaries, but the bulk of its business is in Russia and Russian-speaking countries.
Last week, its board of directors said it will weigh "options to restructure the group's ownership and governance in light of the current geopolitical environment."
It said this could include developing some of its services -- including self-driving technologies, cloud computing and data labelling -- "independently from Russia."
It added that the company "anticipates" that it will "in due course be renamed".
Yandex said it could "divest ownership and control of all other businesses in the Yandex Group", such as advertising, e-commerce and food delivery.
This summer Yandex said it was selling its search engine -- the largest in the Russian-language internet -- to the government-controlled VK group which owns the country's largest social network VKontakte.