The club have refused to comment on the issue, but they have stated that Abramovich is still committed to the team that he has owned since 2003.
Ineos already owns Swiss side Lausanne-Sport.
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Recently, he was a guest at the Blues' training centre, otherwise known as Cobham.
Last month's Sunday Times rich list valued Ratcliffe's wealth at £21.05bn - more than double Abramovich's £9.3bn valuation.
Ratcliffe, who grew up in Manchester, has not previously linked with buying into English football.
In the last couple of years, Abramovich has invested heavily and his money has no doubt helped Chelsea claim five Premier League titles, the FA Cup, the Champions League as well as the Europa League.
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It comes as Mr Abramovich, 51, struggles to renew his United Kingdom visa in the wake of Theresa May's crackdown on rich Russians living in Britain after the poisoning of Sergei Skripal and his daughter Yulia in Salisbury.
Known as a supporter of the United Kingdom leaving the EU, Ratcliffe has had his own run-ins with the British Government.
He is providing financial backing for the Daily Mile Foundation charity and has said that one of his missions is to improve the country's health.
The Mail quote a "friend" of Ratcliffe with regards to any deal involving Chelsea, as saying: "Shelling out £2bn or so would seem like a drop in the ocean for Jim but at the moment he is torn about what to do".
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A spokesman for the company told the Press Association: "We can't comment on market rumour or speculation".