Hidalgo led for the third round in succession at Club de Campo Villa de Madrid on Saturday, shooting 68 to reach 13 under par.
Yet a brillaint six-under par round of 65 moved Rahm, the 2018, 2019 and 2022 champion, to within two strokes.
That was despite playing much of the round without his driver, which suffered damage on the front nine and was only returned to him on the 14th hole after being repaired.
Puig also shot 65 to reach 10 under and set up an all-Spanish final group as he and Hidalgo each seek their maiden European tour title.
Rahm, hailed by Hidalgo after Saturday's round as "the GOAT in this tournament", said: "I'm very excited.
"Hopefully, I can end up with the win but it's definitely going to be a Sunday, atmosphere-wise, that we haven't experienced yet. There's going to be a lot of people watching and it's going to be a lot of fun."
England's Joe Dean holed out from the 12th fairway for his third eagle in two days on his way to a two-under-par 69 that left him tied in fourth, with former US masters champ Patrick Reed on eight under.
Britain's Tyrell Hatton recovered from a bogey-bogey start to post a five-under-par 66 and join compatriot Tommy Fleetwood on five under for the tournament.
The leading Australian is David Micheluzzi, who is in joint-31st place at two under par for the tournament after his moving-day 71. Jason Scrivener is one over after a 70, tied for 52nd spot.
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