Manager Ronald Koeman has made clear that he is not willing uncertainty over the midfielder’s future to drag on beyond this summer.
Academy graduate Barkley, 23, is out of contract at the end of next season, with no new agreement in place.
Barkley produced an outstanding performance as Everton beat Leicester 4-2 in the Premier League on Sunday.
Asked after the match how he would resolve the contract stand-off, Koeman said: “We offer him a new contract and two possibilities, one is sign that contract, if he doesn’t we need to sell the player. It is simple. It is not so difficult in my opinion.”
Barkley combined superbly with Romelu Lukaku, the Premier League’s leading scorer, who struck twice to take his total in the competition for the season to 23.
Lukaku’s future at Everton is also uncertain, with the Belgium striker having recently rejected an offer to extend his contract beyond 2019.
Koeman said: “Even if they don’t perform on the level, we try to keep the best players.
“We spoke a lot about Ross and Rom because they are really important, most of the time the quality of the players can be the difference between Everton and the opponent.
“Today they played really well. We know Rom is a great finisher but Ross played really well between the lines.
“I think he should have scored one but it is what you want to see they played outstanding but the whole team performance was outstanding.”
In an eventful match at Goodison Park, Everton led through midfielder Tom Davies after just 31 seconds, but Islam Slimani’s cool finish and Marc Albrighton’s free-kick put Leicester 2-1 up before the 10-minute mark.
Barkley crossed for Lukaku to power an equaliser midway through the first half, and Phil Jagielka headed in a Kevin Mirallas corner to give Everton a 3-2 half-time advantage.
Lukaku completed the scoring 12 minutes into the second half, drilling in from close range after Jagielka had risen to meet another Mirallas corner, and there could easily have been further goals.
Leicester missed their injured captain Wes Morgan, who will not be ready in time for for the first leg of their Champions League quarter-final against Atletico Madrid on Wednesday.
The centre-back has missed four successive matches with a back problem.
Manager Craig Shakespeare said: “He won’t be ready for Wednesday. It gives opportunities to other people.
“During the run we have had, people have come in and done well. We knew we’d struggle without Wes. There is no timeframe on his return.”
Defeat at Everton ended Shakespeare’s run of six successive wins, five of them in the Premier League, since replacing the sacked Claudio Ranieri in February.
He put out a much-changed side at Goodison Park, resting his first-choice full-backs Danny Simpson and Christian Fuchs, and using winger Riyad Mahrez only as a substitute.
Midfielder Wilfred Ndidi was absent with a minor groin problem, but is expected to be fit for the match in Madrid this week.
Shakespeare added: “The team selection was with the games coming up in mind, on Wednesday, Saturday and Tuesday. We have got a good squad and players have had to be patient.
“The team selection was made to try to get three points. It was more about squad rotation than anything else.
“We played OK in bits. We got off to the worst start possible by conceding a goal after 35 seconds and then showed a lot of character quickly.
“The response was what I was after and we got two quick goals.
“We struggled at times to deal with Everton’s movement and attacking threat, but ultimately we were done by two set plays, which was disappointing.”