Abramovich, 51, is Britain's 13th richest man with a fortune estimated at £9 billion and is closely connected to Vladimir Putin. The refusal to grant a new visa - or at least delay it - will further raise tensions between London and Moscow.
Abramovich's dealings - like those of other oligarchs in the UK - are under scrutiny following the Kremlin-orchestrated nerve agent attack on Sergei Skripal and his daughter Yulia in Salisbury, Wilts, in March.
Abramovich, despite owning at least three homes in London, is currently barred from entering the UK because he no longer has a valid visa.
He missed Chelsea win the FA Cup Final on Saturday although friends have pointed out he watches fewer Chelsea matches than he used to and has little interest in the FA Cup, preferring the Premier League and Champions League.
His old Tier 1 investor visa, which ran for 40 months, expired while he was abroad, it is understood. As a result, Abramovich was forced to make an application for a new visa under tougher rules brought in in 2015, rather than the more simple process of renewing an existing visa.
Theresa May's spokesman, while declining to comment on Abramovich's specific case, said yesterday: "The rules were tightened in 2014-15, that involved amongst other things... new powers were introduced to refuse where there are reasonable grounds to believe the applicant is not in control of the funds; funds were obtained unlawfully or by conduct which would be unlawful in the UK; or the character, conduct and associations of the third party providing the funds in granting is not conducive to public good."
Asked if somebody given a visa under the old system might be barred under the tougher new rules, the spokesman said: "The work is ongoing and I wouldn't pre-empt it but it is a logical conclusion."
The difficulty faced by Abramovich is also privately being blamed on tit-for-tat diplomatic expulsions in the wake of the attempted assassination of the Skripals that has caused delays in processing the Chelsea owner's visa application and those of other wealthy Russians.
Russia closed Britain's St Petersburg consulate and expelled 23 diplomats.
Abramovich's spokesman declined to comment but Mr Putin's spokesman Dmitry Peskov said: "We are witnessing Britain's rather unprecedented anti-Russian mania which is manifested in various ways.
"I have no doubt that such actions won't go unnoticed by investors from other countries. Many countries are working to improve their investment attractiveness and this I believe is a step in the opposite direction."
Boris Johnson, the Foreign Secretary, also waded into the row, suggesting tougher sanctions could be on the way for Russians as part of a crackdown.
Mr Johnson said the impact of American sanctions, which go further than those imposed by the EU, has been "very marked" and told journalists in Buenos Aires he has "noted" their success. Mr Johnson said: "The truth is actually that I think the effect of some of those sanctions [in the US] particularly on some individuals has been very marked and I've noted that."
