The Rocket Man singer extended the offer of talks after speaking at a conference in neighbouring Ukraine, where he met with the country's president Petro Poroshenko to lobby him about lesbian, gay, bisexual and transgender rights.
Sir Elton accused Putin of saying "stupid things" in reference to the president warning gay people travelling to the Winter Olympics to "leave the children in peace".
Russia also passed a law banning homosexual "propaganda" among children in June 2013, sparking an international outcry.
Sir Elton told the BBC: "Give me a break. You are president of Russia, and you go and say stupid things like that?"
Putin's attitude to gay people was "isolating and prejudiced" and "ridiculous", he added.
Sir Elton went on: "I would like to meet (Mr Putin). It's probably pie in the sky ... He may laugh behind my back when he shuts the door, and call me an absolute idiot, but at least I can think I have the conscience to say I tried."
The star thanked Poroshenko for his "support" after meeting with him.
He said on Instagram: "After speaking at the YES Ukraine 2015 Conference today in Kiev, I met with Ukraine President Poroshenko about the importance of legislative changes to support LGBT rights.
"Specifically, I asked him to: 1. Ban discrimination in the workplace based on sexuality and gender identity. 2. Provide criminal penalties for hate crimes.
"We need Ukrainian businessmen and power brokers to create a more inclusive and tolerant society. We need this everywhere!"
Sir Elton has been outspoken on the issue of gay rights before, most recently when, earlier this year, he temporarily boycotted Dolce and Gabbana after the fashion designers called children conceived by IVF "synthetic".
He has two sons, Zachary, four, and Elijah, two, with husband David Furnish.