The Al-Ayyam daily on Friday quoted "informed sources" as saying the letter from Corporal Gilad Shalit to his father Noam "confirms that the soldier is alive, opening the way to a prisoner exchange".
The newspaper said the letter was delivered to Egypt by a delegation from the governing Palestinian movement Hamas, whose armed wing was one of three militant groups that claimed the June 25 raid in which Shalit was seized.
The letter was handed over the same day to the Israeli authorities who delivered it to Shalit's father, Al-Ayyam said.
Speaking to AFP, Noam Shalit refused to confirm or deny the report. No immediate comment was available from the Israeli defence ministry.
The newspaper said Hamas delegates from its exiled politiburo have stayed in Cairo to negotiate via Egyptian mediation the terms of a deal to swap Shalit for Palestinian detainees held by Israel.
Hamas insists that any swap must include jailed Palestinian faction leaders such as Marwan Barghuti from Fatah and Ahmed Saadat from the Popular Front for the Liberation of Palestine, as well as dozens of Hamas officials arrested by Israel after Shalit was captured, the daily said.
Hamas is also demanding the release of Palestinian prisoners serving long prison sentences, and of women and minors, Al-Ayyam added.