The first and most obvious priority for the new Everton manager to sort out is to establish which players will still be around next season. John Stones and Romelu Lukaku, most prominently, have attracted attention and are understood to be impatient for Champions League football, but if the players are sufficiently impressed by Roberto Martínez’s successor they may be persuaded to stay at Goodison.

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What Ronald Koeman will need to find out, and it will not be easy with most Everton players away on international duty for the next few weeks, is whether his leading players are willing to give a new regime a go or whether they are set on leaving. If the latter – bearing in mind that Koeman will have his own ideas about whom he wants to keep and which players he might bring in – it becomes a matter of achieving the best price and finding replacements as quickly as possible.

Stones might be the more replaceable. Although undoubtedly a defender of pedigree and potential, some of his performances last season were less assured than might be expected of a centre-half described a year ago as England’s first choice for the foreseeable future. If José Mourinho is still keen, it might be best to take Manchester United’s money while the player’s value is still high and use the cash to find a defender willing to commit the next few years of his career to Everton.

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Lukaku is slightly different. Reliable goalscorers are a rare commodity in football and Everton have one of the best. The Belgian endured a frustrating start to his career in England, twice being sent out on loan by Chelsea, before becoming Everton’s record signing at £28m, and though still relatively young he is entitled to feel that after five seasons in English football he should by now be winning things rather than scoring to provide a lifeline.

Where Everton would have finished in the table without Lukaku’s goals last season hardly bears thinking about. Koeman will need to persuade him that he can put other players of similar quality around him so he does not feel he is carrying the whole team. It might be possible. If not, Everton will not be short of offers but will find it difficult to get a like-for-like replacement.

Looking at the rest of the team, and beginning at the back, a new goalkeeper would be high on most Everton supporters’ agendas. Tim Howard was great for the club, even if he did stay on a touch too long, and the club really need to find a long-term replacement of the same calibre. Joel Robles still looks like an over-promoted reserve.

If Everton have defensive problems (and they most certainly do), here is the most obvious upgrade. It was not Robles’s fault that so many goals were conceded last season and so many winning positions thrown away, but he is not yet the sort of commanding presence to inspire confidence in the defence in front of him.

Everton actually have a decent defensive line, though Phil Jagielka is not getting any younger and Stones may have to be replaced. At least one centre‑half is likely to be needed, though perhaps more important is working to bring back defensive discipline to the entire team. Martínez was famously woolly about the defensive aspect of the game and he was exactly the same at Wigan before that.

Koeman, as a former defender, will be expected to tighten up that area. He did so at Southampton, building on the foundations laid by Mauricio Pochettino. Outside the top three, only Manchester United conceded fewer goals than Koeman’s team last season.

Like Pochettino, another former defender, Koeman has a reputation for producing teams that are tight at the back but still play attractive football. That is exactly what Everton require. Under Martínez they were beginning to acquire a reputation as a soft touch, a team frequently guilty of over-complication. Most Everton fans could stand seeing their team become a little meaner and more physical, and to that end a driving midfielder would be a useful capture, someone to keep the side moving forward and bring the best out of Ross Barkley.

Bringing the best from him may be the single most pressing issue in the new manager’s in-tray, arguably more important than keeping hold of Lukaku. Barkley is one of Everton’s own, an undoubted talent, yet he did not exactly blossom under Martínez and his England career has been a series of false starts. It would make Koeman’s task easier were Barkley to emerge as one of the stars of Euro 2016.

That could still happen, even though many thought he was lucky to make the squad. Potentially Barkley is the sort of player to build a team around, though for all Martínez’s lavish claims, that potential still has to be fully unlocked. Roy Hodgson has first go in France; after that the challenge passes directly to Koeman.

Beyond that, the challenge facing Everton’s new manager is simple but daunting. All four of last season’s north-west clubs now have different managers than a year ago and the standard is exceptionally high. Everton finished fourth of that quartet by a distance last season and they would like to do much better. Not just to finish fourth by a shorter distance, but perhaps to return to troubling the top four in the division.