A person familiar with the details of a recent trip Evans made to Russia said the offer came at a meeting Evans had with Putin last week at the Kremlin. This person spoke only on condition of anonymity because no public announcement has been made.
Evans did not give an answer at the meeting but said he would consider the offer, the person said.
The Interfax news agency quoted Deputy Prime Minister Alexander Zhukov as telling reporters at a news conference Thursday that the government would in principle welcome the participation of Western executives in Russian companies.
"As for Evans, we'll soon find out," Zhukov said.
Rosneft has previously declined to comment on Evans' possible appointment, directing requests to the Federal Property Agency, which holds the state's 100 percent stake in the company.
No one could be reached at the agency Friday evening.
The Wall Street Journal reported Friday that Rosneft is scheduled to sell as much as 49 percent of its stock in an initial public offering in 2006.
Evans was in Russia last week at the invitation of the American Chamber of Commerce to discuss integrating Russia into the global economy. While in the country, he attended a number of meetings with U.S. executives doing business in Russia and with senior officials of Putin's government.
A week ago, the Russian state-owned gas company OAO Gazprom named former German Prime Minister Gerhard Schroeder as chairman of a $5 billion Russian-controlled venture to build a gas pipeline under the Baltic Sea.