Zaha reportedly turned down a new and improved £125,000-a-week contract last week, hinting that Zaha might not be completely settled at Selhurst Park.
The Yarmolenko funds would not cover the whole Zaha transfer fee, as Crystal Palace have slapped a £70m price tag on their 2018 Player of the Season.
New Everton boss Marco Silva wants to make Zaha a key member of his new attacking line-up but is struggling to convince him to move back up north. At the time of writing, the Toffees are yet to make a single summer signing - though snagging Zaha would certainly be worth the wait.
Zaha moved to Manchester United in 2013 - he was Sir Alex Ferguson's last signing - but got very little game time under David Moyes and was eventually loaned out to Cardiff City before being sold back to Palace in 2015.
Zaha scored nine goals and three assists last season in the Premier League, despite missing five weeks through injury on his way to being named Crystal Palace's Player of the Season for the third season in a row.