At the end of the week the rumours suggest he is now pursuing Israeli citizenship.
Is he about to quit Britain? What is Britain’s problem with him?
Our quest to find out began with an attempt to find the man himself.
He is notoriously private, media shy and heavily protected by an army of bodyguards.
But in this business it’s always worth a try.
The French seaside town of Antibes is the playground for the world’s richest.
And even among them, there are those on an entirely different level.
On the Mediterranean coast, Antibes is a little east of Cannes, a little west of Monaco.
Ferraris and Range Rovers sit outside luxury hotels.
The southern tip of the coastline is dominated by Chateau de la Croe.
Surrounded by trees and a high perimeter wall dotted with cameras and motion sensors, it is only really visible from off shore by boat.
Eighty years ago, the mansion was home to the Duke of Windsor and his American wife, Wallis Simpson.
They retreated here after he abdicated the British throne.
Today, its occupant is a Russian oligarch, a man worth upwards of $11 billion.
Mr Abramovich bought the chateau in 2001, two years before he secured a major foothold in the UK, buying Chelsea Football club.
As the first Russian oligarch in London, he turned the fledgling club around and built up an empire in the British capital.
It now includes his Kensington mansion and a Russian Steel firm, Evraz, which is headquartered in London and, this year, has been the best performing FTSE 100 company, gaining 42%.
A public footpath runs along one of the chateau’s perimeter walls. As we walk along, the cameras move and follow us.
For someone who values his privacy, it’s a wonderful spot.
Moored off shore is his boat, the Eclipse: registered in Bermuda, with a crew of 70, a helipad, a mini submarine.
It is staggeringly large.
Beyond it is another yacht, a little smaller, but still huge.
This was was Mr Abramovich’s too, until he lost a bet and handed it over to a mate.
We approach the Eclipse and politely call out to the staff on board.
As an attempt to find and speak to a Russian billionaire about his visa issues, this was always likely to be futile.
And so it was.
Our first introductions were met with a clear “no”.
But on the radio from our boat to the bridge of the Eclipse we managed a brief convention.
Could we find out, I ask somewhat foolishly, whether Mr Abramovich is aboard and able to speak to us.
“That’s all well received, Sir. At this moment in time we are unable to comment on the location of any guests. That’s unable to comment. Thank you.”
Obviously he wasn’t going to speak to us, but a little bit of his empire up close is a useful hint of the life of an oligarch.
So what’s the British government’s apparent issue? Russian man gets rich, enjoys life, invests in London; plenty have.
Why, as foreign secretary, Boris Johnson said this week, will the UK “continue to tighten the squeeze on some of the oligarchs who surround Putin?”
On the specifics of Mr Abramovich’s visa issue, the UK government has not commented.
Sources close to him, quoted in the Russian media, say there isn’t a problem, just a delay.
But it means that his British footprint is one he cannot tread for the time being.
His Kensington mansion and Stamford Bridge are off limits.
Nigel Gould-Davies is a former British diplomat, former ambassador to Belarus and a close observer of Russian government behaviour.
“The point here is not that anyone is being accused of illicit actions,” Mr Gould-Davis tells me.
“The point is that the sources of Russian oligarchic wealth in London have not hitherto been scrutinised so I think that if anything it’s a matter of holding Russians and hopefully wealth from other countries as well to a more careful standard of transparency and scrutiny than before,” he said.
Mr Gould-Davies was serving in the British Embassy in Moscow when KGB defector Alexander Litvinenko was murdered in London.
“Russian and other wealth in the UK has long been an ethical concern, but we have a dramatic new concern in particular since the events in Salisbury,” he said.
“Russia is presenting a more aggressive threat to the United Kingdom and other Western European countries than it has at any time since the Cold War period.
“Now the Russian state, the Putin regime, depends upon elite networks of power and influence and wealth.
“So the Russian State is on the one hand presenting a threat; on the other hand it’s been sustained and supported by elites who make use of the UK’s financial services to keep their wealth safe.
“That cannot be right.”
On the same day that Mr Abramovich’s visa issues emerged, the British government’s foreign affairs select committee published “Moscow’s Gold: Russian Corruption in the UK”, a report which recommended that the government cracked down on Russian wealth in the UK.
It is something that American banker and author Bill Browder has been calling for ever since he accused the Kremlin of murdering his lawyer.
Sergei Magnitsky died in a Russian prison in 2009.
He had represented Mr Browder, who had been accused of tax fraud by the Kremlin.
He lobbied the US congress to introduce the Magnitsky Act, a law to punish Russian human rights abuses with sanctions.
He has called for the same in Europe and the UK.
“The US has made a very complete list. They made a list of 200 oligarchs close to Vladimir Putin. I think you could operate off of that list and the US sanctioned seven of those individuals. I think the UK as a first step should sanction the same individuals that the US has sanctioned,” Mr Browder said.
Mr Abramovich is not among those sanctioned in the US and has not been accused of any wrongdoing.
“But Mr Browder says all oligarchs close to the Russian president should be targeted.
“Most oligarchs did the deal with Putin and effectively became his business partner.
“And so when you hear about an oligarch worth $13 or $14 billion, probably $7 [billion] of that is Vladimir Putin’s.
“And so, if you want to go after Vladimir Putin, you go after the oligarchs,” Mr Browder said.
Mr Abramovich’s proximity to Putin goes back years.
Well before he acquired Chelsea, he was one of Russia’s most influential businessmen.
He was close to Mr Putin’s predecessor Boris Yeltsin; he helped Mr Putin rise and he benefited from it.
Mr Browder welcomed the decision this week to pass into law the Sanctions and Anti-Money Laundering Bill, which includes the “Magnitsky amendment” allowing the British government to freeze the assets and ban the visas of human rights violators from Russia and elsewhere.
“That’s an absolutely powerful tool that can be used and should be used aggressively against these Russian oligarchs,” he says.
Back on the French Rivera, we watch the comings and goings.
A helicopter lands on the back of the Eclipse.
If there is link between this type of wealth and Mr Putin’s ability to rattle and destabilise the West then a life like this could get a little more tricky.