In recent weeks, Abramovich has experienced problems with gaining a working visa for the United Kingdom and has also decided to postpone plans to build a new stadium.
Doubts also remain over the future of head coach Antonio Conte but according to the Daily Mail, the Russian billionaire has rejected the chance to end his period at Stamford Bridge.
The report claims that Great Britain's richest man Jim Ratcliffe - the boss of multinational petrochemicals giant Ineos - has made a bid for the West Londoners, which was promptly turned down by Abramovich.
Ratcliffe is said to be a fan of Manchester United, but holds a season ticket at Chelsea due to his home being near to the club's ground.
Abramovich bought Chelsea in 2003, with five Premier League titles, five FA Cups, three EFL Cups, the Champions League and the Europa League being won since then.