The 26-year-old Croatia international has been at Everton's Finch Farm training complex to discuss personal terms and undergo a medical.
No details of the fee have been revealed but Rangers were thought to value the player at ?8million, having previously rejected a ?7million package from West Ham.
There were also reports the Merseysiders had been knocked back with a prior offer, so it remains to be seen whether a compromise has now been agreed with the Scottish champions.
A report on the Premier League club's website read: "Everton can confirm a fee has been agreed with Rangers for the transfer of Croatian international Nikica Jelavic."
Everton manager David Moyes has money available following the sale of winger Diniyar Bilyaletdinov to Spartak Moscow at the weekend for a fee believed to be around ?5million.
The Toffees will now hope to conclude the deal well before their home league clash with Manchester City.
Cash-strapped Rangers have been reluctant to part company with their influential forward, who has scored 36 goals in 55 matches since arriving from Rapid Vienna in 2010, but must take the financial aspect into consideration.
The player was also thought to be keen on a move to England having been linked with other clubs including Liverpool and QPR this month.
He would become the third addition to Moyes' squad this month following a loan deal for Landon Donovan and the permanent signing of Darron Gibson.
Source: PA
Source: PA