The Saudi Pro League have sounded out Doucoure's advisers, but have yet to formally approach Everton.
Should he join Al Ettifaq, he would be reunited with ex-Everton teammate Demarai Gray, who made the switch last summer.
Steven Gerrard's side are keen to bring in a new midfielder this month after Henderson's exit was confirmed on Thursday.
The Liverpool legend forced his way out of the Middle East just six months after arriving where he has now linked up with Dutch giants Ajax.
And this has put Doucoure on Al Ettifaq's radar, who is the Toffees' top scorer in the Premier League this season.
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He has found the back of the net on six occasions so far - his most since banging in seven for Watford in the 2017/18 campaign.
It was his goal in the 1-0 win against Bournemouth on the final day of last season that confirmed Everton's safety in the top-flight.
A fan favourite at Goodison Park, the 31-year-old has just 18 months left to run on his contract in Merseyside.
Despite being capped twice for Mali's national team, he decided against playing at the Africa Cup of Nations in order to keep playing for Everton.
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But in a blow for Sean Dyche, he has picked up a hamstring injury, with it remaining to be seen when he will return to action.
Issuing updates on the fitness of Doucoure and Seamsus Coleman after their FA Cup third round replay win over Crystal Palace, Dyche said: "Seamus is hopefully not too long.
"Doucs maybe a little bit longer. It's a hamstring injury.
"We're not sure yet... it's the other leg, nothing to do with the other one, unfortunately - well fortunately in a way.
"Seamus is the same, a minor hamstring. We're hopeful they settle down quickly but it's too early to tell yet."