American Scottie Scheffler, the current best men’s player on the planet, is alongside McIlroy on 10 under.
McIlroy, 35, has put himself into the mix for a maiden Olympic medal after cutting out the mistakes he made before the weekend.
“It was a really solid round of golf,” said McIlroy, who missed out on the podium by finishing tied fourth in his first Games at Tokyo 2020.
“It would have been nice to be able to squeeze a couple more shots out of the back nine.
“But I think tomorrow, depending on what the leaders do, I’m going to have to probably shoot my lowest round of the week to have a chance at a medal. That’s the goal.”
Ireland team-mate Shane Lowry is five under after also carding a 66.
But Britain’s Matt Fitzpatrick had a back nine to forget and later pulled out with a thumb strain.
The Englishman, making his Olympic debut, came in seven over par after the turn – including a quadruple-bogey eight on the treacherous 15th.
He twice found the water circling the green, going on to sign for a 10-over 81, which left him five over in total.