The Egyptian demand came in a "strongly worded letter" from Omar Suleiman, Egypt's powerful intelligence chief, to Khaled Mashaal, the Syrian-based Hamas leader, Palestinian officials and Arab diplomats said on Tuesday.
The message reflects increasing impatience with Hamas by Egypt, which has been mediating for months in an effort to reach a deal on a prisoner swap for the release of Israeli Corporal Gilad Shalit, held by Hamas-allied fighters in Gaza.
The forming of a national unity government with Mahmoud Abbas has also been stalled by Hamas's refusal to form an administration that recognises Israel.
"The leadership has received the Egyptian letter today and is studying it," a Hamas official close to Mashaal said from Damascus.
Shalit, was captured on June 25 outside the Gaza Strip by Palestinian fighters, which was followed by an Israeli military offensive against Gaza.