President Vladimir Putin will step down after eight years in office when his ally and successor Dmitry Medvedev is sworn in on May 7. Putin said he would stay in as premier, but he is widely expected to retain a commanding role.
Putin introduced envoys to the federal districts - seven vast areas comprising several regions each - at the start of his presidency to strengthen Moscow's control over the regions amid post-Soviet chaos.
The envoys, appointed by the president, have enjoyed substantial powers, ensuring regional legislation's compliance with federal laws, and advising the president on candidates for governorships.
Vedomosti said the envoys could now focus on economic issues and would report to the government led by Putin.
"The envoys should be above all efficient managers," the paper quoted a presidential administration official as saying. "Being no experts in economics, many of the current envoys could therefore lose their jobs."
The paper also said, citing a member of the ruling United Russia party, that "envoys will have a new role: they will become the government's envoys overseeing regional branches of ministries and state agencies."
Governors have given mixed reactions to the possible changes.
"We have long needed someone to coordinate the economic activities of Russia's constituent members," said the governor of the northern Russian region of Vologda, Vyacheslav Pozgalev.
"What is good in theory does not always prove useful in practice," Pskov Region Governor Mikhail Kuznetsov said.
The regional development minister, Dmitry Kozak, who is expected to retain the post and oversee envoys' activities, said on Thursday he was unaware of plans to make presidential envoys accountable to the government.