The 28-year-old midfielder suffered the injury during the Blues’ 1-0 defeat to West Ham on Sunday.
Premier League: Rafa Benitez's Everton sceptics grow quieter as flying start propels European dreams
Everton have lost just one of Benitez's first seven Premier League games in charge despite a limited budget due to financial fair-play restrictions and injuries to star strikers Dominic Calvert-Lewin and Richarlison.
Everton remained in eighth place, but the team was only one point behind rival Liverpool and in touching distance of the Champions League places. Arsenal stayed in ninth place.
Premier League: Everton assistant manager confident Dominic Calvert-Lewin will start 'battering' in goals
Calvert-Lewin scored the first goal in Everton's 3-0 win against Championship side Sheffield Wednesday in Sunday's FA Cup fourth-round tie. His 15th goal of the season was his first since early December, although he had missed the previous two matches due to a hamstring injury.
Pep Guardiola's Manchester City have had a patchy start to the season but are now just three points behind leaders Tottenham after a dominant display at the Etihad.
Premier League: Liverpool set new Anfield record with win over Leicester; 10-man Arsenal hold Leeds United
Liverpool have now set a new club record of 64 unbeaten league games at Anfield.
Harry Maguire shines on captaincy debut as England end 35-year winless run against Republic of Ireland
Just 48 hours after the resignation of English Football Association chairman Greg Clarke following his offensive comments about the BAME and LGBT communities, Gareth Southgate's side showed no signs of being distracted by the controversy.
Premier League: Pierre-Emerick Aubameyang's penalty sees Arsenal beat Manchester United; Gareth Bale scores in Spurs' win
Aubameyang ended a five-game goal drought in the league when he stepped up to convert a 69th-minute spot-kick after Paul Pogba fouled Hector Bellerin.
City returned to winning ways after an indifferent start to the season, courtesy of a first-half goal from Raheem Sterling, who condemned Arsenal to their seventh straight defeat in league matches between the two sides.
Premier League: Everton's Dominic Calvert-Lewin credits Carlo Ancelotti's influence as key factor behind England call-up
Calvert-Lewin has scored nine goals already this season, including two hat-tricks, as the Toffees have soared to the top of the Premier League table and progressed to the quarter-finals of the League Cup.
Premier League: Wilfried Zaha nets brace as Crystal Palace stun Manchester United; Arsenal edge West Ham
It was a chastening start to the season for United, who have come under fire from fans frustrated at their relatively modest transfer outlay compared to their title rivals.
Premier League: Dominic Calvert-Lewin nets, James Rodriguez shines as Everton beat Tottenham; Leicester victorious
It was the first time Tottenham coach Jose Mourinho — who has previously managed Chelsea and Manchester United — has lost his opening match of a Premier League season, having been unable to sign a backup option to Kane.
Premier League: Everton boss Carlo Ancelotti avoids touchline ban after confronting referee Chris Kavanagh
Ancelotti was left infuriated after a decision by the Video Assistant Referee to disallow a late Everton winner and received his marching orders from referee Chris Kavanagh for protesting after the final whistle
Premier League: Everton, Manchester United draw in thrilling contest; wily Wolves fight back to beat Tottenham
There was VAR drama in stoppage time in the match between Manchester United and Everton, when the Toffees thought they had scored a winner.
Premier League: Everton's Richarlison, Dominic Calvert-Lewin combine to drub Crystal Palace; Watford, Brighton share spoils
Calvert-Lewin’s 11 goals this season are the best by an English striker at Everton since Andy Johnson in 2006-07
Premier League: Duncan Ferguson makes instant impact as Everton beat Chelsea to move out of relegation zone
Two goals from Dominic Calvert-Lewin after Richarlison’s early opener justified Ferguson decision to shake up the side with an aggressive 4-4-2 formation as the former Everton striker replaced the sacked Marco Silva in the dugout
Premier League: Manchester City win to stay close to leaders Liverpool; Chelsea register first win at home under Frank Lampard
Liverpool, who have won 16 consecutive league games, finished a single point behind champions City in the Premier League last season but are now setting a searing pace as they chase their first top-flight title since 1990.
Premier League: Arsenal blow two-goal lead to draw against Watford; Callum Wilson brace powers Bournemouth to win over Everton
Arsenal played a huge part in their own downfall too as they gifted possession away inside their own box for Tom Cleverley to pull a goal back before David Luiz upended Roberto Pereyra, who slotted home the resulting penalty
Premier League: Chelsea manager Maurizio Sarri says he's 'worried' about team's mentality after defeat to Everton
Maurizio Sarri admitted he is worried about his Chelsea team’s "mentality" after they fell to a 2-0 defeat at Everton that severely damages their hopes of a top-four finish in the Premier League.
Premier League: Everton dent Chelsea's Champions League qualification hopes with comfortable win at Goodison Park
Maurizio Sarri endured a frustrating afternoon as his Chelsea side's second-half collapse saw them suffer a 2-0 Premier League defeat at mid-table Everton.