Ashraf al-Qudra, a spokesman for the Hamas-run health ministry, said a 12-year-old boy was in serious condition after being shot in the head northeast of Gaza City near Jabalia.
Thirteen other Palestinians who had been throwing stones at Israeli soldiers were also wounded when the troops opened fire at them, he added.
An army spokeswoman told AFP "hundreds of Palestinians" had been throwing stones at Israeli soldiers beyond the border fence and attempting to sabotage it.
In the other incident a Palestinian boy, also aged 12, was seriously wounded by Israeli gunfire near the Sufa crossing east of Rafah in southern Gaza, Qudra told AFP.
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A Hamas security officer was moderately wounded when he was shot in the leg, he added.
Hamas security sources told AFP several children, residents of the area, were playing near the border when they were shot at by Israeli soldiers.
An Israeli military spokeswoman said soldiers had identified "an armed suspect" approaching the security fence and fired warning shots before using live ammunition.
Clashes are common on Fridays, with regular protests near the border in support of Gaza farmers who say troops uprooted their trees to create a buffer zone.
Tensions have risen in and around Hamas-ruled Gaza after a year of relative calm, with Palestinian militants firing projectiles at Israel and the Jewish state targeting Hamas facilities and militants.