The former Preston manager has been at the Toffees for more than 11 years since his appointment in March 2002, and took charge of his last home game at the club on Sunday as the Toffees defeated West Ham 2-0.
Everton has been linked with a host of coaches already, from Wigan’s Roberto Martinez to Swansea’s Michael Laudrup, and Kenwright is adamant the club will take its time and consider all their options in the search for Moyes' successor.
"We will not be making a quick decision, we will be taking time," Kenwright told reporters. "We'll get the season over, and then we will hopefully give some indication of where we are going."
Two goals from Kevin Mirallas saw the Toffess record a 2-0 over Sam Allardyce's side, and an 11th clean sheet of the season helped the club secure sixth place and a second successive Premier League finish above Merseyside rival Liverpool.
"It was a great victory," Kenwright said. "It was a great performance by the team, it was a great performance by the crowd and, once again, a great performance by a great manager."