The sentences had been expected to be carried out overnight on Thursday.
"We told the prosecutor we're ready, there's no problem"
Ali Fauzi, a relative of Amrozi and Mukhlas
The three men - Amrozi, 47, his brother Mukhlas, 48, and Imam Samudra, 38 – are expected to die by firing squad at a prison island off the coast of Java,
The twin bomb attacks on Bali nightclubs in October 2002 killed 202 people, many of them foreign tourists including 88 Australians.
Earlier this week protesters marched through Jakarta chanting slogans condemning the trio's executions and praising them as "holy warriors".
Authorities, fearing a massive funeral procession by road, have prepared helicopters to transport the bodies from the high-security prison on Nusakambangan island to their home villages.
Ali Fauzi, a relative of the brothers Amrozi and Mukhlas, said the family has prepared itself "in case the execution takes place".
"They told us not to wait for a letter informing us of the executions because the Bali court has no obligation to send one," he told AFP.
"We told the prosecutor we're ready, there's no problem."
|Police are on high alert as part of tighter security ahead of the execution [EPA]
Security has been boosted across Indonesia amid fears of a backlash from a small minority who support the bombers.
Police are also investigating bomb threats received this week against the US and Australian embassies, and an Internet letter purportedly written by the bombers threatening the life of Susilo Bambang Yudhoyono, the Indonesian president.
The convicted bombers have warned of retribution in a string of authorised media appearances from prison but their lawyers deny they wrote the letter threatening the president.
The condemned men have also said they are eager to die as "martyrs".
Anti-US groups in the country have opposed the execution saying Indonesia was bowing to pressure from foreign countries.
Dozens of Indonesian Muslims protested outside the national human rights commission in Jakarta on Thursday, saying they were against foreign interference in the country's domestic affairs.
"Now, you can see America's intervention on Detachment 88," Zaki Cahyadi, the leader of the 'Indonesia Muslim Soldier' group which organised the gathering, said.
The Indonesian anti-terrorist unit Detachment 88 was credited with capturing leaders of the Jemaah Islamiyah group – allegedly linked to the al-Qaeda network – and its military wing in a series of raids last year.