Alonso and his Aston Martin team were the talk of the paddock after topping for FP2 and FP3, with the expectation that the Silverstone team could at least challenge Mercedes to enter the top three.
The two-time world champion did exactly that in qualifying, taking fifth place behind the four cars from Red Bull and Ferrari and narrowly ahead of Mercedes' George Russell and Lewis Hamilton.
When Motorsport.com asked him if Aston Martin's strong tyre management and race pace means Alonso could take aim at the Ferraris and go for a podium, the Spaniard said it was possible, even if it "felt too good to be true" to even be thinking about it.
"Let's see, yes. I'm laughing because going for a podium in race one... it feels too good to be true," Alonso grinned.
"But for sure, what we saw so far on our car, and also historically Aston Martin has a very good tyre management. So yeah, if we have an opportunity for sure, we will take it."
Amid the hype around the AMR23's performance, this weekend's free practice sessions continued providing evidence that the team built an impressive car but Alonso admitted that fighting with Ferrari and Mercedes "seems a little bit unreal".
"In practice we knew that we were around third or fifth or fourth," the 41-year-old explained.
"It was amazing. The whole weekend has been unreal for us. It was like too good to be true every session and every performance of the car.
"And then in qualifying, our expectation was to be around half a second was from Red Bull, so it was quite right."
Alonso said the team thought Red Bull, Ferrari and Mercedes would prove "untouchable" in 2023 and was quietly just hoping to attack the front of the midfield.
"Again, it feels so strange for me to say that we start P5 and maybe we are aiming for more," he added.
"Because we thought to be honest that the first three teams were untouchable this year with the advantage they had last year, and maybe we try to lead the midfield mix.
"But now we're saying that we start P5 and maybe we fight with the Ferraris for the podium, so this is just incredible.
"So yeah, let's keep working, keep the feet in the ground and execute the race the best way possible.
"No mistakes. Good start good stops. And let's see where we are. But so far, we are living our dream."