Vladimir Putin has mounted a robust defence of Sochi's imminent winter Olympics, insisting that corruption has been negligible despite the $50bn (£30bn) price tag and reassuring gay competitors and spectators that they will not be in danger in Russia.

Less than three weeks from the start of the most controversial winter Games in history, Putin told western interviewers including the BBC's Andrew Marr he was unaware of the systemic corruption that an International Olympic Committee member alleged earlier this month.

"I do not see serious corruption instances for the moment, but there is a problem with overestimation of construction volumes," Putin said. He said some contractors had won tenders due to low bids which they subsequently inflated.

"This price increase, it is sometimes due to contractor's deliberate acts, and sometimes it is due to the fact that the professional valuation of necessary investments, especially in mountain conditions, for a mountain cluster, are not efficient enough."

IOC member Gian Franco Kasper has said that as much as one-third of the $50bn price tag – the largest in the history of the Olympics – has been siphoned off. Others such as Russia's former deputy prime minister and opposition leader Boris Nemtsov have estimated the extent of corruption at $30bn – a figure greater than the GDP of more than half the countries of the world.

Putin said Kasper's words had been taken out of context, and added that no one had proof of corruption. "If anyone has such information, give it to us, please. I repeat once again, we will be grateful. But so far there was nothing but talks," he said.

The Sochi Olympics have been criticised not just by anti-corruption campaigners, but by gay rights advocates incensed at a new Russian law banning gay propaganda. Putin said at the weekend that gay people were welcome in Sochi but should "leave children alone".

In the interview with Marr, he added: "We have recently passed a law prohibiting propaganda, and not of homosexuality only, but of homosexuality and child abuse, child sexual abuse. But this is nothing to do with persecuting individuals for their sexual orientation. There's a world of difference between these things.

"So there's no danger for individuals of this non-traditional sexual orientation, who are planning to come to the Games as visitors, or participants."

"Read our law carefully – and pay attention to its name. It's called a 'ban on the propaganda of paedophilia and homosexuality'. There are countries, including in Europe, where they're debating the possibility of legalising paedophilia. Publicly discussing this, in parliament.

"They can do what they want, but the people of Russia have their own cultural code, their own traditions. It seems to me that the law we adopted doesn't harm anybody. What's more, homosexual people can't feel inferior here, because there is no professional, career or social discrimination against them.

"When they achieve great success, for example Elton John - he's an extraordinary person, a distinguished musician, and millions of our people sincerely love him, regardless of his sexual orientation."

Asked whether athletes or spectators who protest against the law could face action, Putin said: "Protest actions and propaganda are two slightly different things. Similar, but from a legal point of view, protesting against a law is not the same as propaganda for homosexuality or child abuse."