* Moscow mayoral candidate says bans are unconstitutional
* Navalny faces jail but is seeking office in capital
* Putin under fire for law against gay "propaganda"
By Steve Gutterman
MOSCOW, Aug 23 (Reuters) - An opposition leader running for mayor of Moscow said on Friday he would allow gay rights rallies if elected, making clear he believes the authorities' frequent practice of denying them permission is unconstitutional.
Alexei Navalny, who is seeking to boost the opposition and increase his own political clout by challenging the Kremlin-appointed acting mayor, weighed in on an issue that has gained prominence in Russia and drawn the attention of the world.
Western governments have criticised a law against spreading gay "propaganda" among minors, part of the conservative course President Vladimir Putin has charted in his third term following protests against his 13-year rule.
Polls show most Russians support the law, which activists say amounts to a ban on gay rights rallies. It has hurt Putin's image in the United States and Europe and led to calls for a boycott of the 2014 Olympics in February in Sochi, Russia.
For years, Moscow authorities have denied applicants permission to hold gay rights marches, and attempts to demonstrate have often ended in attacks by anti-gay activists who call themselves allies of the Russian Orthodox Church.
Navalny, a leader of the protests that erupted in December 2011 but which have since faded, said he would change that.
"The constitution says that everyone can (demonstrate), and the mayor of Moscow cannot ... forbid this," he said when asked about gay rights rallies in an interview on Ekho Moskvy radio.
"I will adhere to the constitution. The constitution says all people have the right to demonstrate peacefully and unarmed" Navalny," he added.
"If everyone is clothed and marches in an orderly way with their slogans, I do not care at all what their sexual orientation is, where they are going and what they are saying."
Navalny, 37, was sentenced to five years in prison last month for theft after a trial he has said was Putin's revenge for his opposition activities, but he is free pending a ruling on his appeal.
Opinion polls suggest Navalny has no chance of beating acting mayor Sergei Sobyanin, appointed by the Kremlin in 2010, but he believes he might have a chance if Sobyanin falls short of the simple majority needed to avoid a runoff.
In the interview, Navalny also set out a position on labour migrants that may appeal to Muscovites wary of foreigners, saying all people from ex-Soviet states of Central Asia and the Caucasus should be required to obtain visas to enter Russia.
He said performing a regional folk dance on a square outside the Kremlin - something that has stirred ethnic tension in recent years - would be a "violation of public order". (Editing by Andrew Roche)