Mari Alkatiri met members of his ruling Fretilin party in an emergency meeting on Thursday.
Estanislan da Silva, a Fretilin member, said of the prime minister: "I think he is prepared to go."
Many East Timorese say Alkatiri's decision to fire 600 soldiers in March was to blame for clashes and gang warfare that have left at least 30 people dead and sent nearly 150,000 people fleeing their homes.
The violence has been the worst to hit the tiny Asian nation since it voted for independence from Indonesia seven years ago.
The president, Xanana Gusmao - who told Alkatiri in a letter that he wanted him to resign - is to meet the prime minister after Thursday's internal party meeting, da Silva said.
Gusmao told Alkatiri in a letter
that he wanted him to resign
Though Alkatiri appeared ready to step down, the decision would ultimately be up to Fretilin, which holds 55 of 88 seats in parliament, and nominated him for the leadership post, da Silva added.
"If Mari Alkatiri resigned at the moment, the entire government would have to go because he was the one who formed the government. We don't want to leave a vacuum; we will do the best we can."
While Jose Ramos-Horta, the foreign minister and close Gusmao ally, has offered his services as an interim prime minister to resolve the problem, da Silva said that was impossible.
"He has support of the people, but I don't think he has the support of Fretilin to become prime minister," he said.
Fierce gun battles between rival security forces in the capital last month raised fears that the country was plunging into a civil war, but the arrival of a 2,700-strong Australian-led peacekeeping mission several weeks ago helped to ease tensions.
Still, there have been several flare-ups, with arsonists setting fire to five houses and an administrative office in Dili on Wednesday.
Also on Thursday, Rogerio Lobato, the former interior minister, was escorted by Australian troops to a court where he was charged with "attempted revolution, conspiracy, providing state weapons to civilians and association with criminals", said Longuinhos Monteir, the prosecutor-general.
An arrest warrant was issued for Lobato after he was accused of providing weapons to Vincente "Railos" da Concecao, the self-proclaimed leader of a hit squad allegedly commissioned by Alkatiri.
"Lobato turned himself in this morning," Monteir said, adding that he faced 15 years imprisonment if convicted.
Alkatiri has denied involvement in the alleged hit squad, but the Australian current affairs programme Four Corners reported on Monday that the prime minister was told by a police official that civilian militias had been armed with government weapons.