It said they showed Mr Banks met Mr Yakovenko three times - having previously only acknowledged one encounter in 2015 - and made a visit to Moscow in February 2016 in the midst of the referendum campaign.
They were said to have been introduced to Mr Yakovenko by Alexander Udod - one of 23 suspected Russian intelligence officers subsequently ejected from the United Kingdom after the poisoning of Sergei and Yulia Skripal in Salisbury.
"He said he only met the Russian ambassador on one occasion; we now know it was more than that".
But Banks dismissed the claims, telling the paper: "I had two boozy lunches with the Russian ambassador and another cup of tea with him".
Mr Banks and Mr Wigmore also discussed potential business opportunities in Russian Federation, emails reportedly show, including a proposal involving six goldmines.
Banks described calls for a political investigation as "political mischief-making" plotted by pro-Remain MPs.
Some Trump aides are under investigation in the USA over whether they colluded with Moscow and there are also claims that Moscow tried to meddle in Britain's referendum campaign to encourage a Brexit vote.
Banks gave around £9 million to the Leave.EU campaign and Grassroots Out Brexit campaign, and used to fund the UK Independence Party under the leadership of his friend, Nigel Farage.
At a press conference at the conclusion of the G7 summit in the canadian resort of La Malbaie in the province of Quebec to the Prime Minister of the UK Theresa may was asked whether she believed that British security services should investigate the revelation in the article that the head of the government said: "I have not seen the publication in The Sunday Times, but I think that any recognition of the need of the investigation and the competent authorities, it will".
"It's a convenient political witch-hunt, both over Brexit and Trump".
The Sunday Times report was based on emails passed to the paper by the journalist Isabel Oakeshott, who ghostwrote Banks's memoir "The Bad Boys of Brexit" and is now writing a book on Russian attempts to influence British politics.
Banks has also reportedly admitted that he handed over phone numbers for members of Donald Trump's transition team to Russian officials, after meeting with the United States president-elect in November 2016 in NY.
Banks, who last week announced he was pulling out of the Commons inquiry into fake news, accusing the MPs of a "witch hunt", said he would now be giving evidence to the Digital, Culture, Media and Sport Committee as planned on Tuesday.
Banks also told the paper nothing came of the gold mine discussions.
"We didn't profit from any business deals because I never pursued anything", said Mr Banks, who has interests in South African diamond mining.
Nigel Farage revealed that he was once invited to a meeting with Arron Banks and Andy Wigmore at the Russian Ambassador's residence but declined.
Wigmore told the paper: "We never offered any information to him (Mr Yakovenko) or any Russian any details of our (Brexit) campaign".
The announcement came after the Sunday Times newspaper reported that Banks' contact with Russian officials was more extensive than previously acknowledged.