Defenders Phil Jagielka and Michael Keane will be given every chance to be fit for the visit of Burnley on Sunday.
The pair both missed the win over Bournemouth last weekend and Thursday's Europa League draw with Apollon Limassol, with the Toffees skipper also sitting out against Sunderland in the Carabao Cup.
Jagielka (hamstring) and Keane (foot) have both been working hard in order to be fit to face the Clarets, but Koeman revealed a final decision will be made on the defensive duo on Saturday.
“I hope tomorrow I will have some positive news,” said the Blues boss. “We have to make a decision about Michael and Phil over whether they are available for Sunday. They are doubtful.
“First of all, it is important to have everybody available, because those we have out are quality, experienced players. That is what you need in a difficult situation.
“We will put out the best XI possible, with the freshest players, in order to try to win the game.”
Koeman also revealed Wayne Rooney has been passed fit to face Sean Dyche’s men.
The forward injured a hand in the 2-2 draw with Cypriot side Limassol on Thursday night but tests have disclosed no damage.
However, James McCarthy continues to miss out on selection, with the Blues boss suggesting the midfielder was set for a spell with David Unsworth’s Under-23s in order to build up match fitness.
“Wayne had a scan of his hand this morning,” added Koeman. “There is no fracture and he is okay.
“James is not available. He did the first part of the session today but not the rest of it. He needs game time first with the Under-23s. That’s how players need to come back from a long time out injured.”
Jonathan Walters is the newest addition to the Burnley treatment room after he suffered a knee injury which has also ruled him out of Republic of Ireland’s crucial upcoming World Cup qualifiers.
Goalkeeper Tom Heaton misses out after dislocating a shoulder against Crystal Palace earlier this month, so Nick Hope will continue in his place.
Heaton and Walters join Nahki Wells and Dean Marney on the sidelines as they recover from longer-term injuries.
Burnley’s impressive start to the campaign has seen the Clarets rise to ninth place in the Premier League table.
It’s hard to ignore eye-catching away performances such as the 3-2 win at Chelsea on the opening day courtesy of a brace from Welshman Sam Vokes and a wonderstrike from Stephen Ward, and a 1-1 draw at Spurs.
In their most recent trip to Merseyside, Burnley played out a 1-1 draw with Liverpool, a game in which they led through Scott Arfield’s 27th-minute finish.
Another Premier League draw followed the clash with Liverpool, this time against newly-promoted Huddersfield Town.
Overall, the Clarets are unbeaten in the league since their home defeat to West Brom in mid-August, however they did lose on penalties to Championship outfit Leeds United in the Carabao Cup third round.
Everton go into Sunday’s fixture following a disappointing 2-2 draw with Apollon Limassol in the Europa League.
After Adrián Sardinero had opened the scoring, a goal from Wayne Rooney and a first for the Club from Nikola Vlasic had put the Blues in good stead to pick up their first three points in Group E. However, a late Héctor Yuste header secured a share of the spoils for the Cypriot side.
Premier League form, on the other hand, looks to be on the way up for Everton after they secured a much-needed 2-1 victory over Bournemouth last weekend.
Like Thursday night, the Blues fell behind to a Josh King strike, and it could have been 2-0 but for a fine Jordan Pickford save from Jermain Defoe.
Yet, substitutes Tom Davies and Oumar Niasse combined for the latter to level on 77 minutes and the same duo repeated the trick five minutes later to guarantee all three points.
ONE TO WATCH: CHRIS WOOD
Club-record signing Chris Wood has hit the ground running since his £15million summer switch from Leeds United.
The New Zealand international is no stranger to finding the back of the net, having won the Championship Golden Boot with 27 goals in the 2016/17 campaign.
Wood made the perfect start to his Burnley career, scoring on his Premier League debut with a 92nd-minute equaliser to deny Tottenham Hotspur what would have been their first league win at Wembley.
In his second appearance for the Clarets, a home debut against Crystal Palace, the striker finished well in the third minute to give Burnley all three points.
Wood faced his old side, Leeds, in the third round of the Carabao Cup, and kept his cool to score a 89th-minute penalty in a game that eventually finished 2-2. After a goalless period of extra-time, he netted again in the shootout but saw his former employers go through 5-3 on spot-kicks.
The forward began his career at Waikato FC, moving to Hamilton Wanderers before making the switch to England and West Bromwich Albion in 2009.
There were limited opportunities for the then 17-year-old, who was loaned out to Barnsley, Brighton, Birmingham City, Bristol City, Millwall and Leicester City.
A permanent move to Leicester followed in 2013, where he scored 12 times in 52 league appearances. He left the Foxes in 2015 to head back to the Championship with Leeds United and chalked up a very impressive record of 41 goals in 83 league outings before securing his transfer to Burnley.
LAST TIME: EVERTON 3-1 BURNLEY
Everton were victorious in the two sides’ previous meeting, a game in which Romelu Lukaku scored for a ninth successive home game, taking the Blues’ record to eight wins on the bounce at Goodison Park.
Phil Jagielka scored for the third match in a row as he headed home just after half-time. Yet, after being brought down by Joel Robles, Sam Vokes stepped up to level the scoring from the penalty spot.
Another centre-back then scored for Everton, Clarets defender Ben Mee turning the ball into his own net in the 71st minute.
Lukaku got his obligatory goal three minutes later, smashing the ball past Tom Heaton to round off the scoring.
MAN IN THE MIDDLE: JONATHAN MOSS
Sunday’s fixture will be the 150th in the Premier League for Jonathan Moss.
Having qualified as a referee in 1988, Moss made the jump to the top division for his first match in 2011, overseeing Birmingham City’s 2-1 win over Blackpool.
The Yorkshire-based referee officiated Arsenal’s 2015 FA Cup final victory over Aston Villa, in what was his most high-profile fixture to date.
Moss has refereed six times in total this season, including one Everton fixture - the Blues’ 2-0 defeat at Chelsea in August.
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of Saturday’s contest here on evertonfc.com
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