Fleetwood defied miserable conditions at Oak Hill to card a third round of 68.
The former world number one was six over par through three rounds at the US PGA Championship.
The Masters champion bogeyed his first two holes on Saturday.
McIlroy carded a second round of 69 for a halfway total of level par to lie just five off the lead.
McIlroy is five off the lead shared by Scottie Scheffler, Corey Conners and Viktor Hovland.
The Englishman is looking to add to his 2013 US Open title.
Block was one off the clubhouse lead after a second successive level-par 70 at Oak Hill.
The head pro at Arroyo Trabuco Golf Club in California carded a second consecutive 70 for a halfway total of level par.
A frost delay of almost two hours meant the first round had not been completed on schedule on Thursday.
The 29-year-old leads the way at Oak Hill on four under par.