Bourdais suffered pelvic injuries and a broken right hip after a horrific crash during Saturday's qualifying session at Indianapolis Motor Speedway.
"Dale Coyne, the owner of Bourdais" team, told Motorsport.com: "[On the] first day after the crash he was saying he wanted to be back for Sonoma [the IndyCar season finale].
"My thoughts are with Sébastien and (wife) Claire Bourdais", Davison said. "When they said that we have only one shot today that created a little bit of stress for everyone on the team, but I had the vehicle to be confident and to push all four laps".
The team was led by Takuma Sato, who had the second-fastest time Saturday with a four-lap average speed of 230.382 miles per hour. It's all "ifs", but I think he'd have been eighth or ninth in Phoenix without that accident, third in the Grand Prix (of Indianapolis) without the engine failure, and a front row or even a good shot at the pole yesterday.
Beyond the Indy 500, questions remain as to how long Bourdais will be out of the vehicle, and who will replace him at other events.
6 facts about Memorial Day and how to celebrate it
They passed the Uniform Monday Holiday Act, which set the last Monday in May as Memorial Day and to declare it a federal holiday. A Memorial Day parade will take place Monday morning, followed by Memorial Day exercises at the Four Seasons Centre.
"I'm feeling better and better, physically and also with the vehicle as well", Alonso said.
Asked about morale within the team, Coyne said: "Everyone's sad because we all know how good this year could have been".
He won the season-opening race at St Petersburg, Florida, and had the fastest time in Friday's practice (NZT Saturday).
It was the ninth-best speed of the day, but Jones starts 11th because he didn't qualify for the "Fast Nine" final shootout with his Saturday speed. "Now we just wait for him to recover". He is a four-time Champ Car World Series champion.
Scott Dixon driving a Honda-powered vehicle for Chip Ganassi Racing, the 10th driver out, held the top spot for most of the day with a four-lap average speed of 230.333 miles per hour. However, he underwent a successful surgery and now plans to return at some point.
He did not take part in the second qualifying day on Sunday and will make his debut in the No. 18 auto on Monday.