Worth an estimated £9.3 billion, according to the Sunday Times, Abramovich would become the richest person in Israel and would be exempt from the country's taxes on income earned overseas for 10 years, as well as not being required to declare the source of that income.
His visa application came against a backdrop of deteriorating diplomatic relations between London and Moscow over the poisoning of Russian former double agent Sergei Skripal and his daughter in March.
Abramovich has been a regular visitor to Israel and is reported to have bought a former hotel in Tel Aviv, close to the Mediterranean coast.
It is not clear if the United Kingdom chose to reject his application permanently but a source familiar with the matter told the Daily Telegraph that the renewal process was taking an unusually long time.
Joe Skipper Reuters Abramovich's little boat
Abramovich would have to explain the source of his wealth to receive a new British visa, according to reports.
He is set to move to Tel Aviv after buying a property there.
The British government has declined to comment on his case.
Israeli passport holders can enter Britain without a visa for short stays. In Israel, meanwhile, he will be exempt from paying taxes and declaring income for ten years, under a controversial law that police reportedly say was pushed by Benjamin Netanyahu to benefit his billionaire friends in exchange for gifts. It is not known whether a foreign passport was issued to the Russian businessman. He said: "The truth is actually that I think the effect of some of those sanctions, particularly on some individuals, has been very marked and I've noted that, but we have our own systems and our own approach and we have to do it in accordance with the law and accordance with the evidence".
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President Donald Trump went on Twitter on Saturday to thank Seaman "for his heroic act in saving so many precious young lives". Students were bused to Noblesville High School gym, where their families could retrieve them. "You'll be proud of them, too".