Speaking on September 10 to members of the Valdai International Discussion Club, Shaimiyev said greater influence would be in the interest of Russia's federal subjects.
"I don't see any other way of development," RFE/RL's Tatar-Bashkir Service quoted Shaimiyev as saying.
The Valdai International Discussion Club is a think tank, created by Russia, which brings together top Russian officials, politicians, economists, and foreign political analysts.
Some 100 delegates from 12 countries are taking part in the group's fourth session, held this year in Tatarstan's capital of Kazan.
Shaimiyev also said that the opinions of local assemblies should be decisive in appointing regional heads. He said it was "wrong" that Russian law made it possible to disband regional assemblies if they fail to confirm candidates nominated by the Russian president.
"I think the right to disband law-making bodies is wrong. I have said this before, and am saying it again," Shaimiyev said. "I am sure this will change with time. A law-making body is a law-making body. It has been elected by the entire population, therefore it has the last say."
In 2005, Russian President Vladimir Putin replaced the direct election of regional executive-branch heads with a system under which regional legislatures confirm candidates who are nominated by the president.
Almost all the heads of Russia's 89 regions have already been appointed this way.
RFE/RL Russia Report
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