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Cherif Chekatt was killed in the Neudorf/Meinau area of the city after a police operation was launched around 2100 hrs (2000 GMT) on Thursday about 2 kilometres from where he launched his attack on Tuesday.
Chekatt was killed after firing on police officers, who returned fire, one of the sources said.
Reuters reporters near the scene heard three to four gunshots after a huge police operation with armed forces from the BRI and RAID units. A police helicopter had been circling overhead.
The death toll from Tuesday's attack rose to three as police on Thursday combed the city in the east of France for a second day and manned checkpoints on the German border.
Police issued a wanted poster in multiple languages for Chekatt, who was the main suspect in the attack and who had been on a watchlist as a potential security threat.
Authorities say the 29-year-old was known to have developed radical religious views while in jail.
Earlier in the day armed and masked police swooped on Strasbourg neighbourhoods with elite RAID officers fanning out across three locations in late afternoon, including the area where Chekatt was last seen.
The Paris prosecutor's office said Chekatt's parents and two brothers were being held in custody.
Two of his sisters in Paris were also questioned on Thursday and one of their homes was searched, a judicial source said.