There is no evidence that Abramovich has done anything wrong, but the United Kingdom has scrutinized Russian businesspeople and diplomats more carefully since the poisoning of former Russian spy Sergei Skripal and his daughter Yulia in Salisbury, England, in March.
An official at Nativ, a department of Prime Minister Benjamin Netanyahu's office dealing with Jewish communities in the ex-Soviet Union, said Abramovich had applied for an immigration visa at Israel's embassy in Moscow, the station said earlier on its website.
Abramovich owns the Varsano boutique hotel in Tel Aviv's Neve Tzedek neighborhood, which he bought for NIS 100 million in 2015 and is converting into his Israeli home.
In Britain, Mr. Abramovich had an investor visa - valid for three years and four months and normally extended every two years - which expired at the end of April.
The story was headlined: 'Roman Abramovich has become a new immigrant to Israel'. The newspaper also reports that he is now Israel's richest citizen. The Kremlin denies any involvement.
It is understood the tycoon regards such matters as personal.
Joe Skipper Reuters Abramovich's little boat
He qualifies for Israeli citizenship through his Jewish heritage.
An Israeli passport will allow Mr Abramovich to enter Britain for up to six months, but he won't be able to work there without first receiving a work visa.
The move has not been confirmed by Abramovich's Moscow-based spokesman John Mann, while Chelsea declined a request from ESPN FC to comment.
Israel grants citizenship to any Jew wishing to move there, and a passport can be issued immediately.
Local reports suggest that he will take up residency in Tel Aviv, having bought a property from actress Gal Gadot.
Fulham back in the Premier League
In a big game like today we need those decisions to go our way and we need to play better than we did in the first half. On the half-hour mark, Fredericks contested with Grealish for the ball, losing out to the midfielder who was fouled.