Mourinho has reportedly made the Bosnian international is number one transfer priority at Stamford Bridge and is willing to let Manchester City take a free run at signing Cavani this summer to make it happen.
The Blues had also been linked with the Uruguayan international, but his £60m price tag has led Mourinho to look elsewhere. And by stepping aside, Mourinho hopes City will play ball in negotiations over a Dzeko transfer.
City are thought to want around £20m for Dzeko, who's fallen behind Carlos Tevez and Sergio Aguero at the Etihad Stadium.
The 27-year-old former-Wolfsburg striker was linked with Real Madrid, then managed by Mourinho, when he signed for City in the 2011 January transfer window.
Mourinho often prefers to play with a tall, central striker, and reportedly doesn't feel Fernando Torres or Demba Ba can fulfil the role.
Romelu Lukaku's future at the club remains uncertain, and the young Belgian international may be sent out on loan again following his successful stint at West Brom last term.
Radamel Falcao had been strongly linked with the Stamford Bridge side, but his move to French club Monaco ended the Blues hopes of landing the Colombian.
With Robert Lewandowski likely to join Bayern Munich, the pool of Champions League quality strikers is diminishing rapidly.
But the report claims Mourinho hopes his plan to drop out of the Cavani race will help him seal the deal for Dzeko.