"This is the third invitation to the government of Bangladesh ... if [it] fails to establish the Islamic law or try to arrest any member of Jamatul Mujahidin Bangladesh ... [we] will take direct action," the group said in an internet statement.
"Everybody is the enemy of Islam who wants to launch democracy as an institutional form. Therefore we invite the ruling party and also the opposition to initiate the rule of Islam within a short time in Bangladesh," the statement said.
Two people were killed and about 100 injured as hundreds of home-made bombs exploded simultaneously across the country of 140 million last Wednesday.
Bangladesh police said on Tuesday they had made "substantial progress" in their investigation into last week's nationwide wave of bombings, as they questioned a prominent Muslim leader arrested while boarding a flight to Dubai.
Intelligence officers were interrogating Moulana Fariduddin Masud, a senior member of the Jamiat Ulamaye Islam, a body of prominent Muslim clerics. He was arrested at Zia airport on Monday.
Radicals have targeted public
rallies, concerts and shrines
"We're interrogating him about his departure and frequent travels abroad," Inspector-General of Police Abdul Kaiyum said.
The national police chief declined to comment on newspaper reports that police were investigating whether Masud, who heads several Islamic charities, funded another Muslim group - the outlawed Jamayetul Mujahidin - which the authorities have named as the prime suspect in the blasts.
But Kaiyum said authorities had made "substantial progress" in their investigation after more police raids on madrassas or Islamic seminaries after the 434 blasts that killed two people and injured more than 100.
Kaiyum said 150 people from different towns and cities across the country had been arrested and some had confessed to planting bombs.
A senior security official, speaking on condition of anonymity, said: "Moulana Masud was arrested in connection with the bombings."
The official said foreign travel bans had been imposed on more than 100 people, including Masud. He did not identify the other people or say why the ban was imposed.
Bangladeshi police are hunting
for two controversial clerics
Leaflets of the Jamayetul Mujahidin, calling for the introduction of Islamic law in the world's third most populous Muslim country, were left at the sites of the explosions.
Police are hunting the leader of Jamayetul Mujahidin, Shaikh Abdur Rahman, saying he is believed to have masterminded the attacks. They say many of the people interrogated have told them that Rahman orchestrated the bombings.
Masud is a former director of the Islamic Foundation, the government body which finances the nation's mosques. He was being interrogated by intelligence officials at a special unit set up in the city, a security official said.
Wave of bombings
Police are also looking for Siddiqul Islam, also known as Bangla Bhai, the alleged leader of the Jagrata Muslim Janata Bangladesh, which police believe is a sister organisation of Jamayetul Mujahidin.
Both groups, which want secular Bangladesh to adopt strict Islamic law, were banned in February after being linked to a wave of bombings at charities, Muslim shrines and musical events.
"We're now pretty sure Abdur Rahman was the mastermind of the attack and everything will unravel if we can arrest him," a security source said on Monday, describing the leader as "about 50 and fluent in English".