Four-time major winner McIlroy needs only to win the Masters green jacket to complete a career Grand Slam and arrived after seven top-10 finishes to start the year, including a win at last month's Players Championship.
"I think that Rory is playing probably better than anybody is playing right now," Nicklaus said.
"If I were in Rory's position, I would be looking at trying to win the Masters, not trying to finish a Grand Slam. To win the Masters, that's enough to worry about."
McIlroy could become only the sixth player to complete the career Slam, joining a group that includes Player, Nicklaus, Tiger Woods, Ben Hogan and Gene Sarazen.
"It will be fantastic for him to win," Player said. "If I had to pick somebody I would like to see win, Rory, because it would join the five of us and it would give golf a big boost.
"Whether he wins or not this week, he has the best swing in the field without a question."
Nicklaus says extra focus from sports and golf television and social media has made the task of completing a career Slam harder than in his day.
"When I won the British Open in '66, I don't even recall anybody even mentioning it," Nicklaus said, noting his wife Barbara would often toss the newspaper before he could read about himself.
"It's probably undue pressure put on fellows. If you're smart, you probably shouldn't read all that stuff. You just move on and go do your thing and prepare yourself and go play."
Nicklaus, a 79-year-old US golf icon who has won a record 18 major titles, and 83-year-old South African Player, whose nine major wins include three Masters titles, have handled Augusta National honorary starter duties as a duo since Arnold Palmer died in 2016.
WATCH: Nicklaus, Player kick off 83rd Masters
Player will be among those cheering on 14-time major winner Woods, a contender at last year's Open and PGA Championship in his return campaign after spinal fusion surgery.
"Everybody wants to see Tiger Woods win more majors because he moves the needle like nobody playing golf today," Player said. "Everybody likes to see a man make a great comeback.
"I don't think there's ever been a man that had as much talent. If Tiger never had the problems he had, which were numerous, he would have won at least 20, 21 majors."
Woods, who hasn't won a major since the 2008 US Open, is chasing a fifth green jacket at age 43 and told Player: "I'm not finished yet."
"To hear it come from him, that's encouraging," Player said. "Can he win five majors to beat Jack? I don't think so."
Nicklaus and Player praised last weekend's inaugural Augusta National Women's Amateur but noted the club's long-time men-only membership policy that was in place until 2012.
"There have been some chairmen in the past that were absolutely reluctant to change," Player said. "I was quite severely criticized when Tiger and I were two of the players that said they should have a few women as members."
"Now these ladies playing here - a few guys would turn in their grave because a lot of them stood up in front of these guys and said, 'I will never allow a woman to be a member.' So they will be watching that, from wherever they are."
Nicklaus was impressed with the event, particularly winner Jennifer Kupcho and her closing five-under par 67 that saw her shoot five-under over the last six holes.
"Can't believe that girl shot 67," Nicklaus said. "For somebody to have not played this golf course and to have made those putts, having only seen the golf course the day before, it's pretty sensational."