A nationwide survey by the Washington-based International Foundation for Election Systems (IFES) showed Susilo Bambang Yudhoyono with 61.2% to Megawati's 29.3% with the rest undecided or keeping their choice secret.
Another poll by the independent Indonesian Survey Institute showed Susilo with 57%, Megawati with 31.5% and the rest undecided or unwilling to say.
The IFES poll was conducted between 2 and 9 September, the latter the day when a bomb blast outside the Australian embassy in Jakarta killed nine people and wounded more than 180.
The other survey was done from 10 to 12 September.
Megawati defends record
IFES officials said economic issues rather than security has consistently been the primary concerns of voters, and that Thursday's blast was unlikely to change that.
But as three days of official campaigning began on Tuesday, both candidates took part in a televised discussion in which they outlined broad policies, with fighting terrorism topping traditional issues such as the economy and jobs.
"With your help, Insh Allah (God willing) we will achieve victory on 20 September"
Megawati defended her government's record on security, saying the latest attack, the third in two years in Indonesia, could not easily have been prevented.
"Terrorism can take place anywhere. Let us take the example of the United States ... the Twin Towers September 11 incident could not be prevented and it took place," she said.
Susilo - Megawait's former senior political and security minister - also focused on tackling terrorism, saying the emphasis should be placed on giving intelligence agencies a preventative role.
"What is clear is that their ability for early detection must be enhanced. But what is also clear is that the intelligence [agency] does not work alone," he said, adding that security forces must enhance cooperation.
President Megawati is lagging
behind, say pollsters
The candidates' stilted televised appearances, the first of three late night panel discussions, were criticised by local media as an anti-climactic end to a colourful year of electioneering for Indonesia, where full democracy is a new concept.
With polls such as the latest IFES survey indicating personality was more important than policies for voters, Susilo - who took the lead in a 5 July first round election - remains the favourite.
Influential 'little people'
But in recent weeks, Megawati has been trying to shed her taciturn image by engaging in a number of public appearances in an effort to prevent her key ''little people'' supporters from defecting to her populist rival.
Megawati's support base has come
from rural and poorer districts
About 2000 children, who eke out a living begging and singing on the dirty side roads and dilapidated buses of Jakarta, met Megawati on Wednesday at her residence before taking to the streets to sing her praises, the state Antara news agency said.
"With your help, Insh Allah (God willing) we will achieve victory on 20 September," Megawati said after handing out T-shirts and pins to the children.