Kinnear said that he hoped to have secured a new full-back already, and that David Moyes will have to make do without one for the Palace clash.
It comes amid Everton's strong links to Palace's Daniel Munoz, who would cost upwards of £30 million pounds.
The Toffees face stiff competition for his signature, with Tottenham Hotspur among the clubs interested.
Speaking to BBC Radio Merseyside, Kinnear said: “I don’t think we are in a bad position on that.
"The players we thought we had the option of are still available for us, I don’t think they will be there for Crystal Palace, which was the objective.
“I understand the fans' frustrations; I understand how they see it as being critical.
“Even the Prime Minister chipped in to give us some helpful direction. So it’s reached a level of national importance.
“So we are working hard to bring one in and I’m sure we will. I’m sure it will be an improvement, and it will round off the squad nicely.”
While 30-year-old Munoz weighs up his options, it's likely that Jake O'Brien will start at full-back for Everton this weekend, as Moyes' men look to make a positive start to the new campaign.