The bill, submitted by Medvedev to the Duma last month, will likely come up for a crucial second reading next month, Duma Deputy Viktor Ilyukhin said.
"We need at least 30 days to work on amendments to the bill," said Ilyukhin, a senior Communist Party official and deputy head of the Duma's Constitution and State Affairs Committee.
Under the bill, which includes amendments to 25 current laws, federal ministers -- including the prime minister and his deputies -- and their family members would be required to make public declarations of their assets and income.
The legislation would also require state officials to blow the whistle on their superiors in corruption cases. It would also force officials who leave government jobs to obtain permission from their former bosses before taking jobs with companies they dealt with while in office.
The bill would also force officials to hand over any gifts worth more than 5,000 rubles ($185) to the state.
Ilyukhin said his committee would push for amendments to the bill passed Friday that would allow for the confiscation of corrupt officials' assets.
Furthermore, the bill should expand the type of relatives of state officials required to declare their income and assets to include parents, siblings and adult children, Ilyukhin said.
Under the bill passed Friday, only an official's immediate family members submit declarations.
"So far, these are the main amendments we would like to have, but there will be more," Ilyukhin said. "We are closely studying the bill."
After his election in March, Medvedev declared war on corruption, saying it was undermining economic growth and the country at large.
The anti-corruption bill has been the focal point of his campaign, though critics have derided the legislation as misguided and full of loopholes.
Three-fourths of Russians said the government's battle against corruption had either yielded only "negligible" results or that there had been no improvement in the situation, according to a VTsIOM poll released in September.
Ten percent of respondents said the situation was getting worse, while 7 percent said they had seen some progress.
Corruption in Russia is at its worst level in eight years, according to an annual survey released in September by Transparency International, a global corruption watchdog.
In the survey, the Berlin-based organization ranked Russia 147th in the world in terms of transparency and rule of law -- on par with Bangladesh, Kenya and Syria.
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