Mr Ban is holding talks with Gen Than Shwe in the remote administrative capital Naypyidaw.
Ms Suu Kyi, a Nobel Peace laureate, has spent much of the past two decades in prison or under house arrest.
Her trial on charges of breaking the terms of her house arrest, has been adjourned for another week.
'Ruse'
The UN secretary general said in his opening statement to the junta leader: "I appreciate your commitment to move your country forward... I would like to contribute, to work together, for peace and prosperity," reported AFP news agency.
Ms Suu Kyi's lawyers have been appealing against the judge's ban on testimony from three defence witnesses. One additional defence witness will be allowed to testify.
"The Supreme Court did not send the case files to the lower court, so the case has been adjourned until July 10," said her lawyer Nyan Win. He added Ms Suu Kyi had expressed surprise at the further delay.
The trial of 64-year-old Ms Suu Kyi has caused outrage around the world.
Critics of Burma's military government have dismissed it as a ruse to keep the opposition leader locked up until after next year's election.
The BBC's Laura Trevelyan, in Naypyidaw, says the secretary general is also likely to urge that next year's elections in Burma have some legitimacy.
Our correspondent also says Mr Ban is said to believe he has a rapport with Gen Than Shwe.
The government called national polls in May 1990 which Ms Suu Kyi's National League for Democracy (NLD) won convincingly, but the junta refused to hand over control, and has remained in power ever since.
The BBC's Jonathan Head says the failure of many previous UN missions to Burma has at least lowered expectations of this one.
But Mr Ban needs to come away with something more than a few token prisoner releases if he is to avoid charges that his visit is merely being used by Burma's military rulers to boost their legitimacy, our correspondent says.
'Fear and dislike'
Mr Ban has called for the immediate release of all Burma's political prisoners, thought to number about 2,100.
"Through my meetings... I will convey exactly what the international community expects and wishes [regarding] the way they want to see changes in Myanmar [Burma]," he told reporters in Singapore before his departure.
It is not clear if Mr Ban will be allowed to meet Ms Suu Kyi - he would be the first UN secretary general to do so if that happens.
This is Mr Ban's first visit to Burma since he persuaded the junta in May 2008 to accept international aid in the wake of Cyclone Nargis, which killed nearly 140,000 people.
Critics say that despite a number of visits to the isolated country by UN special envoy Ibrahim Gambari and UN humanitarian co-ordinator John Holmes, there is little sign that the UN has wrung any other concessions from the regime.
But some analysts speculate that Mr Ban may have been given an indication by the generals - perhaps via Mr Gambari, who last week paved the way for this trip - that the visit could harbour some kind of positive result.
Mr Ban is due to meet members of her NLD and other opposition activists.
"I will try to meet with representatives of all registered political parties including Aung San Suu Kyi, that's my hope," he said, adding that he would ask for the meeting at his talks with Gen Than Shwe.
Our correspondent adds Mr Ban will also try to persuade the reclusive general to re-start a dialogue with Ms Suu Kyi, a woman the leader is said to fear and dislike.
UN special envoy Ibrahim Gambari "clearly conveyed" his wish to meet her when he visited Burma last week, he said.
Ms Suu Kyi was transferred from house arrest to prison in May after an American man swam to her lake-side house. She faces up to five years in jail if convicted.
The NLD leader led a revolt against Burmese dictator Gen Ne Win in 1988, calling for peaceful democratic reform and free elections.
But her movement was brutally suppressed by the army, which seized power in a coup on 18 September 1988.
Human Rights Watch said Mr Ban should not accept the return of Ms Suu Kyi to house arrest as a sign of a successful visit.
"Time and again, the UN has politely requested Aung San Suu Kyi's release, but her 'release' back to house arrest would be a huge failure," said executive director Kenneth Roth.
