* Russian companies hold annual meetings by end-June
* Kremlin's list confirms Sechin to lose Rosneft chair
* Kremlin orders 2011-13 privatisation schedule by May 15
(Adds order to strip officials of chairmanships by Oct. 1)
By Melissa Akin and Alexei Anishchuk
MOSCOW, April 2 (Reuters) - President Dmitry Medvedev gave Prime Minister Vladimir Putin a July 1 deadline on Saturday to begin the process of removing Russian ministers from state firm boards, a move which could clip the wings of top Putin allies.
Medvedev's order set no deadline to remove the ministers from boards, but said they must be stripped of chairmanships by Oct. 1. Among those who could lose posts are Putin's powerful deputy Igor Sechin, chairman of the board of state oil company Rosneft (ROSN.MM).
The call for Sechin and other Putin allies to relinquish posts as board members of state companies has stoked speculation the president may be challenging his mentor ahead of the March 2012 presidential election.
Medvedev and Putin have jointly ruled Russia since Medvedev succeeded Putin as president in 2008, with Putin widely seen as the more powerful of the two men.
They have said they will decide together which of them will run for president next year. Analysts give conflicting views over how serious a perceived split between them is. [nLDE72U07U]
Russian companies hold their annual meetings by the end of June. The July 1 deadline left it unclear whether Medvedev would force out powerful bureaucrats from their corporate posts during shareholder meetings in May-June, or merely start the paperwork.
"This shows he intends to bring this idea to life," Konstantin Simonov, director of the National Energy Security Fund in Moscow, said. "Before his term expires he can make some real trouble for his bureaucratic rivals."
As chairman of top Russian oil producer Rosneft (ROSN.MM), Sechin presided over a landmark share swap and Arctic drilling deal with BP (BP.L). Rosneft holds its annual meeting in the southern city of Krasnodar on June 10.
The list of officials to give up seats, published on the Kremlin website (www.kremlin.ru) as an appendix to a sheaf of orders on measures to improve the investment climate, goes further than initial expectations. [nLDE7310AI] Under the order, Finance Minister Alexei Kudrin will lose his seat on the board of No. 2 Russian bank VTB (VTBR.MM) as well as his seat on the board of diamond miner Alrosa.
When Medvedev proposed the changes on Wednesday evening, some analysts thought monopolies like Alrosa would be exempt.
Medvedev also demanded by May 15 a final list of companies to be privatised this year and next, and asked for improvements in corporate disclosure to minority shareholders.
The order to remove powerful bureaucrats from posts at firms sent a strong signal, but Medvedev has floated vaguer versions of the same idea since he was steered into the top job by Putin in 2008, so the test is how far he succeeds.
Many investors and analysts believe Medvedev and Putin are enacting an elaborate charade to give the impression of competition in a tightly controlled political system.
"I have always said that the Putin-Medvedev conflict is blown out of proportion," Simonov said. "But the conflicts between the people below them -- they are real."
(Reporting by Melissa Akin; Editing by Ron Askew and Peter Graff)