Everton, who managed to preserve their Premier League status on the final day of the season, also activated the option to extend midfielder Abdoulaye Doucoure's contract until the end of June 2024.
Everton said backup keeper Begovic rejected a new deal offered by the Merseyside club, with the Bosnian penning a farewell message to the fans earlier this week.
"We wish all of them the best with the next chapters in their careers," Everton director of football Kevin Thelwell said in a statement.
"We have also offered new contracts to players and we will continue talking with them as we look to build a competitive squad for (manager) Sean Dyche and his staff for the new season."
Coleman, whose contract was also expiring this month, saw his season end early in May after a knee injury in a draw with Leicester City which required the 34-year-old to be taken off on a stretcher.
Dyche said later that Coleman, who has been with the club since 2009, had an operation on the knee and would be fit for next season.
