Garcia, playing before an enthusiastic home fan base at Club de Campo del Mediterraneo in Castellon, Spain, shot 6-under-par 65, with one bogey.
Finch, who was tied for second when the round began, shot 5-under 66, with two bogeys.
Kjeldsen, the first-round leader, ended his day with the second of two bogeys at the par-4 18th, which led to a round of 4-under 67.
The trio's 36-hole total is 11-under 131.
"This week is great," Garcia said. "It's nice to be in this position as there is a little bit of pressure playing on my home course."
At 9-under 133, five others, including last week's winner at the Portugal Masters, Alvaro Quiros, share fourth place.
Among those missing the even-par 142 cut were Oliver Fisher (142), Steve Webster (144), and Graeme Storm (150).