They say the proposal will help improve government management in the regions.
If adopted in its present form, it will give regional governors the option of abolishing the post of mayor in over 80 of Russia's provincial capitals, after holding a referendum.
In regions where the mayor is axed, governors would take over some of the mayoral powers.
They would also appoint lower bureaucrats to assume the remaining mayoral functions -- but without the title of mayor, and without a public vote on who manages the city.
The measures would not apply to the mayoral posts in Moscow and St. Petersburg, which have special status.
The proposal has drawn much praise from governors, many of whom are involved in fierce squabbling with local mayors for power and control over local tax revenues.
Mikhail Kuznetsov, the governor of the Pskov Oblast in northwest Russia, says such a law would end the infighting.
"Relations between federal subjects and city mayors are marked with a number of systemic problems. The fact is that city mayors and regional administrations are jointly responsible for many areas of activity and public life," Kuznetsov says.
"The appointment of people to fulfill the mayoral function, rather than the election of mayors, is the simplest solution to this problem. At any rate, it's better than the current situation."
Not surprisingly, mayors are crying foul.
The mayor of the northern city of Arkhangelsk, Aleksandr Donskoi, says the plan is unfair.
"This draft law is flawed," Donskoi says. "What deputies are proposing is to enable federal officials to shift responsibility for their failures to municipal heads. I don't like the fact that only those acting as mayors of [provincial] capitals would become appointed. I consider that if they are appointed, then all representatives of the executive branch should be appointed too, including heads of districts, of villages, and so forth."
Political analysts are also strongly critical of the bill. Many see it as a move initiated by the Kremlin to secure loyal city officials ahead of the State Duma elections next year and the presidential elections in 2008.
The popular elections of governors were slashed two years ago, effectively making governors Kremlin appointees.
Moscow could now be interested in reining in city mayors, who have influence over millions of rural voters as well as a large portion of the revenues that flow into federal coffers.
Provincial capital mayors have, until now, tended to be more outspoken than appointed governors.
Nizhny Novgorod Mayor Vadim Bulavinov has complained about the centralization of power, for instance. Arkhangelsk's mayor, Aleksandr Donskoi, announced in October that he plans to run for president in 2008.
In a State Duma dominated by Unified Russia deputies, the draft stands a good chance of becoming law.
But Yury Korgunyuk, a political analyst with the Indem think tank in Moscow, says scrapping mayors may actually not be in the Kremlin's interest.
"In almost every region of the Russian Federation, all the conflicts that take place are between the governor and the capital city's mayor. This [draft law] fits perfectly into the general tendency of consolidating the power vertical," Korgunyuk says.
"But the big question is: does the [federal] center need this? The center has a strong interest in maintaining this conflict at the regional level. If city mayors are placed under the authority of governors, then governors will become the sole leaders in regions."
The first reading was initially slated for today, but the vote was postponed at the last minute.
Some observers believe the bill's supporters are seeking to avoid controversy ahead of the meeting of the Congress of Local and Regional Authorities, a branch of the Strasbourg-based Council of Europe, in Moscow on November 13-15.
This body has expressed "serious doubts" about the draft law.
Other say the delay reflects the Kremlin's own hesitation.
Significantly, Boris Gryzlov, the leader of Unified Russia and a staunch ally of President Vladimir Putin, condemned the draft law last week. Mayors, he said, must continue to be elected.
Demonstrators speak with local politicians in Butovo about the destruction of a local forest in July 2006 (RFE/RL)
IS RUSSIAN DEMOCRACY MANAGING? Russian President Vladimir Putin has said Western powers seek to pressure Russia under the pretext of concern over its democratic development. He has said Russia is ready to listen to "well-intentioned criticism," but will not allow anyone to interfere in its internal affairs. The Kremlin has been criticized for stifling political oppostion, increasing central control over the media, and cracking down on the work on nongovernmental organizations.