The 26-year-old will have just 18 months left on his current deal when the January transfer window opens, and the club may hope to tie him down after his recent return to full fitness.
Calvert-Lewin has regained his place as Everton's first-choice centre-forward after competing for a place with Beto on his return from injury.
Now, Football Insider claims that Everton are prepared to offer the striker a new contract in the near future and will not consider any offers for him from the Saudi Pro League.
A recent return from injury has seen Calvert-Lewin flourish in an Everton side whose form has picked up greatly in the last few months.
Without the 10-point penalty imposed on them by the Premier League, Everton would be just two points off the top half, after picking up 13 points from a possible 21.
Sean Dyche came in for praise for holding back on reintroducing Calvert-Lewin to the Everton first team, as he recovered from years of setbacks.
A lack of viable alternatives means Calvert-Lewin was often rushed back without giving time for his muscular issues to fully heal under Frank Lampard, and in the early stages of Dyche's reign.
However, despite playing 45 minutes in Everton's final pre-season fixture before the start of the new season, Calvert-Lewin was not risked in the league opener versus Fulham.
A gruesome facial injury suffered in his first game back against Aston Villa saw him ruled out for another month, but that layoff and September's international break provided more time for the striker to regain full fitness.
Since then, Calvert-Lewin has featured in every game, starting the last eight, including two in the EFL Cup, scoring four goals along the way.
That has kept Everton's summer signing Beto on the bench, but having strength-in-depth will be crucial for a side that has been handicapped by misfiring attackers in recent years.
Stop-gap options like Neal Maupay and Salomon Rondon both failed to impress when thrust into the first team to cover for Calvert-Lewin's injuries, scoring just one goal each in a combined 56 Premier League appearances.
After another international break where Calvert-Lewin has been able to manage his workload, he should start up top for the Toffees when Manchester United visit Goodison Park on Sunday.