Allardyce will no doubt be looking to strengthen in the January transfer window, and both Watford striker Deeney and Sevilla midfielder N’Zonzi are said to be high on his list.
However he also soon hopes to have all his absent players back in action so he can fully assess the Everton squad he inherited before the window opens.
“I think getting the players who are injured, getting them fit again, is a main priority before everybody starts talking about the January window,” Allardyce said.
Summer signings Davy Klaassen and Sandro Ramirez have struggled to adapt to the Premier League, while the likes of Ross Barkley, Michael Keane and Leighton Baines are also out injured.
“I have to say, when I’ve looked at this season’s history of games played, players getting injured, it’s no wonder they have struggled in all honesty,” said Allardyce.
“The number of fixtures early on in the season, the amount of games played in the Europa Cup and then the League Cup and the Premier League and injuries on top of that.
“The new players haven’t been able, for me – especially the foreign players – to settle in and show what they can really do yet.
“So hopefully we can get more out of them and they can be more determined to show more than they have already because we’ll need them going into the Christmas and New Year period, which is going to be another hell of a battle.”
A goal apiece from James Tomkins and Luka Milivojevic handed Crystal Palace a crucial 2-0 win over Huddersfield at the John Smith's Stadium on Saturday evening.
Manchester United bounced back from their Champions League exit as they beat Brighton 2-0 at Old Trafford on Saturday night to reach the FA Cup semi-finals.
Tottenham reached the FA Cup semi-finals after they cruised to a 3-0 victory over Swansea in their quarter-final clash at the Liberty Stadium on Saturday.
Premier League basement club West Brom squandered the lead as they suffered a 2-1 defeat to Bournemouth at the Vitality Stadium on Saturday.
Chesterfield boss Jack Lester felt his side were unlucky not to claim all three points after their 1-1 draw at Cheltenham.
New Swindon manager Phil Brown revelled in his side's victory at Cambridge in his first game in charge.