Doucoure has been ever-present for the Toffees this season, and played the full 90 minutes for a third successive game in their win over West Ham United on Sunday.
The 30-year-old has been one of the most productive players in the Everton side since Sean Dyche's arrival, scoring eight goals and setting up a further three since the former Burnley boss arrived in January.
The Malian midfielder was also responsible for arguably Everton's most important Premier League goal in years, when he fired in the winner against Bournemouth on the final day of last season to seal top-flight survival.
However, Doucoure's contract at Goodison expires at the end of the season, leaving him open to agree a potential pre-contract agreement with another club come January.
Fabrizio Romano claimed last week that the club are in talks over a deal which will see Doucoure sign on until the summer of 2025, with an option of extending that until 2026.
Doucoure was out of contract this summer, but Everton activated a clause in his deal that saw it extend for another 12 months.
However, Shamoon Hafez states that the player has offers on the table from around Europe should a deal not be agreed.
Speaking on the topic, manager Dyche said: "We are ongoing with all the players and their agents, keeping in the loop with them."
Hafez does admit that the likeliest outcome is that Doucoure will sign a new deal at the Goodison though, thanks to his close relationship with the club.
Doucoure has endured fluctuating feelings during his spell at Everton since joining from Watford in 2020.
After being signed alongside James Rodriguez and Allan by Carlo Ancelotti, the Italian manager utilised Doucoure in a much different role to the one he enjoys now - playing deeper in midfield as opposed to just off the central striker.
Doucoure was given more freedom to attack in Rafael Benitez's short tenure at the club, but he was then virtually frozen out by Frank Lampard and barely featured during his 12 months at the club.
There is no doubting Doucoure's importance to the side now though, as Dyche has started him in all of Everton's 10 Premier League games so far this season, with the win on Sunday taking them to 15th, five points clear of trouble.