Ireland internationals Stephen Ward and Jeff Hendrick combined for the only goal of the game as Burnley consigned Everton to a fourth loss of the season.
Koeman's cause has not been helped by an unkind fixture schedule that has featured not only mid-summer Europa qualifying games that disrupted pre-season but also testing meetings with the two Manchester clubs, Chelsea and Tottenham.
The former Inter boss is the leading candidate to take over at the Goodison Park if Koeman faces sack. Good shape, good energy and a will and desire to see the game through - I think that mixture is important.
"It's only seven games in, there is a long way to go".
Should Everton axe Koeman, they should do everything they can to lure over Ancelotti, a three time Champions League victor who has won league titles in England, Italy, Germany and France.
"It is not nice to get two weeks of an worldwide break if you lose". It's going to be an interesting few weeks around Goodison Park.
Yet, by the time Rooney bounded on in the 63rd minute at the tip of a midfield diamond, he found his team-mates floundering with Koeman's reshuffle unable to remedy familiar failings.
Sean Dyche's Burnley side will be hoping to continue their great form away from home during their trip to Merseyside.
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