He can make the 33-car field only through a six-car shootout on Sunday that will determine the race's final three drivers.
"We have another chance to be in the race," Alonso, who is seeking to make his second Indy 500 field, told IndyCar TV.
The Spaniard twice got inside the top 30 but both times was bumped outside the mark, with JR Hildebrand and Graham Rahal the final two drivers to seal qualifying spots.
Britain's Pippa Mann will be the only woman in the race, qualifying 30th in 227.244 mph.
Race positions 10-30 were determined during the day-long qualifying with Saturday's nine fastest drivers set to return on Sunday to decide pole.
Spencer Pigot, driving the Ed Carpenter Racing Chevrolet, led qualifying with a four-lap average of 230.083 mph,
He was followed by 2018 winner Will Power of Australia (230.081) and Simon Pagenaud of France (229.854).
The troubles for Alonso, whose top four-lap average was 227.224 mph, were exacerbated by a punctured tire in his first qualifying run.
"That didn't help," Alonso told reporters. "But, obviously, our performance has been quite bad all week. Quite poor."
The Spaniard crashed his Chevrolet in practice on Wednesday and missed nearly two full days of practice while a backup car was prepared. He returned on Friday.
Alonso said he was worried the McLaren team was "not ready for the challenge."
"We've been slow. You see Juncos Racing crashing yesterday and being ready at 6 o'clock. That's impressive," he said. "For us, we've been a little bit slow on everything."
Alonso is bidding to join Graham Hill as the only drivers to achieve the 'Triple Crown of Motorsport' with an Indy 500 victory.
He previously won the Monaco Grand Prix and Le Mans 24 Hours race. He also won two Formula One drivers' titles.
(Reporting by Gene Cherry in Raleigh, North Carolina; Editing by Greg Stutchbury)
