"The Israeli Mossad assassinated him in Malta on his way back from Libya to Damascus," said one of the sources, who declined to be identified.
There was no immediate comment from Israel.
The Islamic Jihad movement, which opposes the PLO peace deal with Israel, has carried out attacks against Israelis, including a suicide attack on a bus stop near Tel Aviv that killed more than 20 people earlier this year.
In Malta on Friday police identified the dead man as Ibrahim Shawesh. Islamic Jihad sources in the Gaza Strip said this was an alias used by Shqaqi, who was 43, and the founder and spiritual leader of the Islamic Jihad.
In Damascus, a senior Palestinian official confirmed that the man killed in Malta was Shqaqi. The Maltese police described the murder as "a professional job".
Maltese newspapers and radio said he was shot in the head at point blank range by a man in black in the resort of Sliema,hours after arriving in Malta.
Shqaqi formed Islamic Jihad after his arrival in Gaza in the early 1980s. The Israelis arrested him in 1985 and expelled him to Lebanon in 1988.
A medical doctor, Shqaqi was born in the teeming Rafah refugee camp in Gaza.
Islamic Jihad is the second largest Muslim militant movement among Palestinians after Hamas, the Islamic resistance movement.Reuse content