The report suggests, with Frank Lampard, Ashley Cole and John Terry coming towards the end of the Stamford Bridge careers, Chelsea manager Roberto Di Matteo is keen to draft in another dressing room leader.
Toure has emerged as one of the Premier League's top centre midfielders since his transfer from Barcelona, and the Ivorian powerhouse was a big reason why City stormed back to win their first league title last year.
And the 29-year-old declared over the summer, through his agent, that he would like a new challenge, leading to speculation he could be set to quit the Etihad Stadium.
The Star reports Chelsea are likely to face competition from Europe's top clubs for his signature, including Bayern Munich and Paris St Germain, although Barcelona could be tempted to resign their former player at the right price.
The intense competition for Toure has reportedly led Abramovich to green-light a huge £300,000-a-week contract offer to persuade the former-Barca star to reject overtures from elsewhere and instead join the European champions.
That huge wage would put Toure in line with football's current highest earner, Anzhi Makhachkala forward Samuel Eto'o. Toure currently earns £150,000-a-week as a basic salary, rising to £220,000-a-week including bonuses.
But Manchester City are likely to be extremely anxious to avoid the Ivorian leaving the Etihad Stadium, let alone joining a Premier League title rival. City signed Jack Rodwell and Javi Garcia over the summer, but Toure is still Roberto Mancini's premier centre midfielder.
Chelsea have started the season in excellent form, but could have room for another box-to-box midfielder. Frank Lampard's contract expires at the end of the season, and should he depart that would leave Ramires and John Obi Mikel as the Stamford Bridge club's senior players in the middle of the park.