A senior officer from Mari Alkatiri's office confirmed the receipt of a letter from Gusmao and said the prime minister would "most probably" resign.
The Portuguese Lusa news agency reported that in the letter sent on Tuesday, Gusmao said he had lost confidence in the prime minister after watching a documentary accusing Alkatiri of supplying weapons to civilians.
The Australian Broadcasting Corporation programme "makes serious accusations on your involvement in the distribution of arms to civilians", Lusa quoted the letter as saying.
"After seeing the Four Corners programme, which enormously shocked me," the letter continues, "there is nothing left for me to do except give you the choice: Either you resign, or, after hearing the State Council, I will fire you, because you no longer deserve my confidence."
Gusmao said in the letter that he expected an answer from Alkatiri "before 1700 today, June 20, 2006".
Politicians back call
On Wednesday, several senior members of East Timor's ruling Fretilin party also urged Alkatiri to resign, saying it would clear the way to resolving a weeks-long difficult situation.
The party members want Gusmao to lead a transitional government until polls due next year, said Jorge Teme, a member of Fretilin's 596-member central committee.
Alkatiri is accused of supplying
weapons to civilians
East Timor sank into chaos after Alkatiri in April fired 600 soldiers, nearly half the tiny nation's army, who had complained of discrimination because they came from the country's west.
Twenty-one people died last month as sporadic battles between rival soldiers and police descended into gang clashes.
The government appealed for foreign help, and more than 2000 combat-ready foreign peacekeepers, mainly from Australia, are deployed in the capital Dili where the local police are no longer functioning.
Rogerio Lobato, the former interior minister, was put under house arrest on Tuesday as an inquiry was conducted into accusations he had distributed weapons to the militias during last month's violence.
A UN administration ran East Timor after it voted for independence from Indonesia in 1999, until 2002.