Abramovich had faced unusually long delays in renewing his UK visa, which had expired, sources told the Reuters news agency. The British government has declined to comment on his case.
Israel's interior ministry confirmed the offer of citizenship to a local television station on Monday.
According to the Ynet website, which belongs to Israel's biggest selling daily, Yedioth Ahronoth, Abramovich flew to Tel Aviv on Monday and received documents confirming his status as an Israeli citizen. He has reportedly bought a property in the city.
"Roman Abramovich arrived at the Israeli embassy in Moscow like any other person," an Israeli government spokesperson was quoted as saying. "He filed a request to receive an immigration permit, his documents were checked according to the law of return, and he was indeed found eligible." Abramovich's spokesman did not respond to a request for comment.
Israel grants citizenship to any Jewish person wishing to move there and a passport can be issued immediately. Israeli passport holders can enter the UK without a visa for short stays, although they require visas to work.
Abramovich, who is Jewish, has been a regular visitor to Israel. Ynet said he had bought a former hotel in Tel Aviv, close to the Mediterranean coast.
The Times of Israel said that, as a new citizen, Abramovich would be exempt from paying tax in Israel for 10 years and would not need to explain the sources of his wealth in that time.
It is believed the British government would have required some financial disclosures if Abramovich's UK visa was to be renewed.
There have also been suggestions that Abramovich fell foul of tighter regulations imposed in 2015. His failure to secure a new tier 1 visa, which allows anyone who invests more than £2m in the British economy to stay for 40 months, comes at a time of heightened tensions between the Russian and UK governments.
Relations have been strained since the poisoning of the former Russian double-agent Sergei Skripal and his daughter Yulia in Salisbury in March, an act Britain has blamed on Russia but in which the Kremlin denies any involvement.
The UK's foreign secretary, Boris Johnson, hinted last week that the UK could take action against Russian oligarchs in the wake of the poisoning of the Skripals.
Johnson said it would be "totally wrong" to discuss the case of Abramovich, who is a friend of Vladimir Putin. But he added: "There is a broader question about what the UK can do to crack down on people close to Putin who may have illicit or ill-gotten wealth. As you know, we have statute that allows us to use unexplained wealth orders against them, but we live under the rule of law so whatever happens must be done in accordance with the rules and the law."
Johnson added that the more stringent approach to Russian oligarchs adopted by the US had been effective. 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."
There is no suggestion that Abramovich has been involved in any wrongdoing.
The Russian foreign ministry had criticised the British government over the delays in his visa application process, saying: "It's obvious that, for the British government, both visas and the regulation of the media sphere are instruments of politics."
Last week, Roman Borisovich, a Russian anti-corruption campaigner, said the Russian ambassador to the UK's suggestion that oligarchs who have similar problems could take their cases to court was unlikely.
"The last thing these guys want is to have a public hearing in the UK. They would have to answer questions like 'how did you make your money' and 'are you the real owner of your assets'? Plus 'are you a conduit for the Russian regime or an instrument for meddling in other countries?'"
hitting "friends of putin"?, following "the own system"? the "law"? "evidence"? (no suggestion Abramovich has been involved in any wrongdoing...).
Wow, that IS a new record in my bullshit-bingo book.