While near-neighbours Liverpool have already tied up four deals, the Toffees have yet to make a move, although enquiries have gone in for two players. Kenwright feels the real business will start once Martinez - currently in the United States working as a television pundit for the Confederations Cup - returns this weekend ahead of the transfer window officially opening on Monday.
"There is no concern at all, not in the slightest," said the chairman, speaking after the club's first Annual General Meeting in five years.
"He has been away and you will always find July 1 onwards - when the current contracts end - is when it will start.
"Liverpool have had a sale (Andy Carroll to West Ham) in the transfer market, have got some money in and are spending some money.
"We have two enquiries going on for two players and if he (Martinez) has any concern I haven't felt one ounce of it. He is back at the weekend raring to go."
Kenwright said pivotal for Martinez is holding on to the players he has already and supplementing them with a couple of new arrivals, while seeking to promote from within the youth system.
"He hasn't met the squad yet. It is difficult for me talking for Roberto but he thinks the squad is fantastic," he added.
"He wants to hold on to it but he also wants to bring in two or three players in the positions where we need them. The first step is to get the players back and get to know them and bring in two or three.
"His absolute passion is youth and he wants to get those players who are there or thereabouts in the under-21s moving forward to play for the first team."