One of the meetings was a November 2016 lunch three days after Banks visited Donald Trump, then president-elect, along with Farage and another prominent Brexit campaigner, Andy Wigmore.
Banks and Wigmore were introduced to Yakovenko by Alexander Udod, a suspected Russian spy who was expelled from the United Kingdom after the poisoning of Sergei and Yulia Skripal.
The paper said that Mr Banks and Leave.EU director of comunications Andy Wigmore had also had lunch with the ambassador in November 2016 - just three days after they and the former UKIP leader Nigel Farage had met Donald Trump in NY following his election victory.
Banks dismissed the leaked emails on Sunday evening, telling the Guardian: "I am not involved in Russian espionage".
Mr Banks and Mr Wigmore also discussed potential business opportunities in Russian Federation, emails reportedly show, including a proposal involving six goldmines.
The emails were passed to The Sunday Times by journalist Isabel Oakeshott, Banks' ghostwriter on The Bad Boys of Brexit, who is now writing a book with the Tory peer Lord Ashcroft on Russia's use of "hybrid warfare" to influence British politics.
Any suggestion that Leave.EU received financial help from Russian Federation was "complete, absolute garbage", he said, positioning himself as the victim of a "witch-hunt". The group operated separately from the official Vote Leave campaign. Mr Banks said that he will tell the lawmakers about his connections with many countries during the campaign.
Asked about the report at the G7 summit in Canada today, Prime Minister Theresa May said: "I am sure that if there are any allegations that need investigation the proper authorities will do that".
"It's a convenient political witch-hunt, both over Brexit and Trump".
Details of the alleged meetings contained in emails published by the Sunday Times, come amid questions about possible Russian influence over the UK's vote to leave the European Union.
She came forward after her email accounts were hacked, according to the newspaper.
He told the paper he had also disclosed details of his contacts with the Russians to United States officials.
Arron Banks, the biggest financial donor to the Brexit campaign, has claimed that he is the victim of a "witch-hunt" after facing calls to explain his links with the Kremlin.
Labour MP Stephen Kinnock told Sky News he will be sending a letter on Monday to the police urging them to launch a criminal investigation into links between Mr Banks and the Russian government.
"We didn't profit from any business deals because I never pursued anything", said Mr Banks, who has interests in South African diamond mining.
However on Monday Mr Banks tweeted: "I have Russian visas in my passport clearly stamped, no visit in Feb 2016".
The paper quoted Andy Wigmore, a close associate of Banks who was present at the meeting, as saying that they did not offer "any information to (the ambassador) or any Russian any details of our campaign".
Senior cabinet minister David Lidington told the BBC today that the allegations were "serious" and should be looked into by the authorities.
He added: "I didn't realise they met him subsequently".
Kabul blast: At least 12 killed in Afghan ministry attack
Pakistan has always been accused of supporting the Taliban and providing safe haven to its leaders - charges Islamabad denies. Arif Noori, a spokesman for the provincial governor, says women and children were among those killed in the explosion.
Vase found in a shoebox sells for €16 million
The price was more than 20 times the estimate of 500,000 euros to 700,000 euros auctioneers Sotheby's had put on the item. A Sotheby's spokeswoman said: "They knew it had some value but nothing like that, nor that it was from the Qian dynasty".
Man holds two people hostage in Paris
A pregnant woman also managed to escape and was being treated for shock by ambulance services on the scene. A police source said the attacker had doused the remaining two hostages in petrol.