Ban pushed Than Shwe to release all political prisoners and hold talks with the detained Nobel Peace laureate Suu Kyi when they met in the bunker-like capital Naypyidaw on the first day of his visit on Friday.
But Than Shwe stalled on the UN chief's request to meet Suu Kyi , who is on trial for breaching the terms of her house arrest and faces up to five years in jail.
There was no indication, however, about whether Than Shwe would accept the request after Saturday's meeting.
"I told him that I wanted to meet her in person. He told me that she is on trial but I told him this is my proposal, this is important and I am waiting for their consideration and reply," Ban said on Friday.
"I am leaving (Saturday), so logically speaking I am waiting for a reply before my departure," he added.
'Considerable resistance'
Ban said he had also sought the release of more than 2,000 political prisoners that the UN says are held in Myanmar, including Suu Kyi, ahead of elections promised by the ruling generals for 2010.
UN officials travelling with Ban later said there had been a "very lively exchange of views" after Ban proposed a five-point agenda for reforms.
There was "considerable resistance" to the proposals, including the establishment of a UN "good offices" bureau in Yangon to provide a permanent structure for Ban and his special UN envoy to Myanmar, Ibrahim Gambari.
Ban's visit had been considered diplomatically risky because of its timing during Suu Kyi's trial, and rights groups warned that it would be considered a major failure unless he managed to win her freedom.
'Show trial'
The 64-year-old opposition leader was transferred from house arrest to Yangon's notorious Insein prison in May to face trial after an American man swam uninvited to her lakeside house.
She has been in detention for most of the past two decades since the junta refused to recognise her party's victory in the country's last elections in 1990.
She appeared in court in Yangon on Friday but the trial was adjourned for a week because the judges had not received an earlier judgment barring two defence witnesses, her National League for Democracy said.
The case has sparked international outrage, with Barack Obama, the US president, calling it a "show trial", and a host of world leaders and celebrities calling for her release.
The visit is Ban's first to Myanmar since he persuaded the junta to accept international aid following Cyclone Nargis in May 2008, which killed around 138,000 people.
Myanmar, formerly known as Burma, has been ruled by the military since 1962.