The sources said Ezzedine Ghora, 23, was hit by two bullets in his arm and chest, and died before reaching hospital.
Israeli police, in a statement on the incident, said a border police force was in the refugee camp of the city of Jenin to carry out arrests.
"Upon leaving the camp, the force identified a suspect trying to throw an explosive device at them," it said.
One of the border police shot him and wounded him, the statement said, adding that the device probably exploded near the suspect.
In an online statement attributed to the Jenin branch of Gaza-based Hamas, the Palestinian religious group appeared to claim Ghora as a member.
"Hamas bids farewell to its heroic martyr... and urges the Palestinian Authority and its security forces to stop their security coordination with Israel,” it said, referring to an unpopular agreement under the 1993 Oslo autonomy accords.
A local Hamas source told AFP Ghora was an "activist" of the movement in Jenin, but he could not confirm his involvement in any of its military activity.
Hamas's military wing paid tribute to Ghora but did not name him as a member.
Jenin's refugee camp is the scene of frequent clashes between Palestinians and Israeli troops, with the army carrying out regular raids to arrest suspected militants, including during a mass crackdown on Hamas in June 2014.
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