The bomb blast in northern Israel severely wounded an Israeli Arab man.
The military now says forces shot the suspected attacker dead in a car close to the border, and that he had an explosive belt, a rifle and a pistol.
It is investigating whether Lebanon’s Hezbollah group is involved, in what would mark a significant escalation.
Israel and Hezbollah, which is backed by Iran, fought a month-long war in 2006.
Monday’s bomb attack took place at Megiddo Junction – a major road intersection about 57km (35 miles) south of the Lebanese border.
A 21-year-old resident of the nearby village of Salem, who was named by Israeli media as Sherif al-Din, suffered shrapnel wounds to his head and body.
Medical officials said he was still in a serious condition at a hospital in the city of Haifa on Wednesday morning.
On Wednesday afternoon, the Israel Defense Forces (IDF) announced that the suspected attacker was spotted in a car outside the border town of Shlomi a few hours after Monday’s bombing.
Officers from the Shin Bet security service and Yamam police special forces unit shot and killed the suspect during an attempt to arrest him, the IDF said, explaining that they believed he presented a “clear danger” to them.
He was “found in possession of weapons, including an explosive belt ready to be activated”, according to the IDF.
The driver of the car, who is an Israeli citizen, was arrested and is still being questioned.
The IDF said its current assumption was that the suspect crossed from Lebanon into Israel either on Saturday or Sunday, carried out the bomb attack, and then tried to get back by asking the Israeli to drive him to the border.
It also said it was “examining a possibility of the Hezbollah terrorist organisation being involved”.
The IDF declined to confirm whether it knew the identity of the attacker. But if this was a cross border attack by a Hezbollah militant it would be the first known incident of its kind in years, and could have the potential to spark a further serious escalation.
More than 1,000 Lebanese and 159 Israelis were killed in the 2006 conflict between Israel and Hezbollah, which erupted after Hezbollah militants launched a cross-border attack in which eight Israeli soldiers were killed and two others kidnapped.
The two sides have not been involved in major combat since then, although there have been sporadic clashes and in 2018 Israel said it had discovered a network of “attack tunnels” dug under the border by Hezbollah.
UN peacekeepers are deployed along the so-called Blue Line that demarcates the border between Lebanon and Israel, which remain formally at war with each other.