Moyes took charge of the Toffees at Goodison Park on Sunday for the final time, just days after it was announced he would be succeeding Sir Alex Ferguson in the Old Trafford hotseat in the summer.
He was given a rousing reception by the crowd, appreciative of the work he has done over the past 11 years, though speculation had already begun regarding a possible replacement.
The likes of Wigan boss Roberto Martinez, Celtic manager Neil Lennon, Swansea's Michael Laudrup and Cardiff's Malky Mackay have all been linked with the Toffees job in recent days.
Kenwright, though, insists the club will take their time over the appointment, with no decision until after the final Premier League game of the season against Chelsea at Stamford Bridge next Sunday.
"We will not be making a quick decision, we will be taking time," said Kenwright. "We'll get the season over and then we will hopefully give some indication of where we are going."
Everton gave their exiting manager a fitting send-off with a 2-0 over West Ham that ensures they will finish the season in sixth place, above neighbours Liverpool for the second season in succession.
Kenwright was full of praise for the players, his manager and the supporters following an emotional day at Goodison Park.
"It was a great victory, a great performance," he added. "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."