The British government has declined to comment on his case.
A spokesman for Abramovich also declined to comment on the report.
Mr Abramovich, who is Jewish, flew to Tel Aviv on Monday and obtained documents confirming his status as an Israeli citizen, the Israeli news site Ynet reported.
Immigration officials told the BBC that he was interviewed last week at the Israeli embassy in Moscow.
The root cause of his United Kingdom visa complications comes from his refusal to disclose the sources of his income.
Joe Skipper Reuters Abramovich's little boat
Abramovich, who is worth an estimated £8.6 billion ($11.4 billion), exercised his right under Israel's Law of Return, which allows Jews from anywhere in the world to gain Israeli citizenship.
Relations between Moscow and London have been strained since the poisoning of former Russian double-agent Sergei Skripal in Britain in March, an act Britain has blamed on Russia but in which the Kremlin denies any involvement. According to the newspaper, the businessman will live in a manor in Tel-Aviv's prestigious neighborhood Neve Tsedek he bought from Wonder Woman star Gal Gadot.
The Sunday Times Rich List put Mr Abramovich's wealth at £9.3bn, and he is now the richest man in Israel. Israeli passport holders can enter Britain without a visa for short stays, which would mean that Abramovich would be able to visit England, though he will still need a visa to work there.
Having amassed a huge and controversial personal fortune through dealings in the Russian oil industry in the 1990s, Abramovich acquired Chelsea in June 2003 and has invested well in excess of £1 billion in the club, bankrolling a 15-year run that has brought 15 major trophies to Stamford Bridge.