Ban said: "I told him that I wanted to meet her in person.
"[Than Shwe] 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."
Suu Kyi's trial, on charges of violating her house arrest, had been due to resume on Friday, but officials announced at the last minute it had again been delayed until July 10.
The trial had already been adjourned for more than a month.
The 64-year-old opposition leader and Nobel peace prize winner has been held in jail or under house arrest for 13 of the past 19 years.
|Ban has said he will press concerns directly with Myanmar's ruling generals [Reuters]
"This is going to be, I know, a very difficult mission," Ban told reporters en route to Myanmar from Singapore.
"But at the same time I know that to bring changes to Myanmar, political conciliation and democratisation, we need to do our best."
He said he would "convey the concerns of the international community, about the slow pace of political conciliation and democratisation process".
Earlier, Ban had said his talks would focus on "three of the most important issues for the future of Myanmar".
He said the issues include:
- gaining the release of all political prisoners;
- resumption of dialogue between the military government and its opposition;
- and creating conditions for credible elections.
Speaking to Al Jazeera, Trevor Wilson, Australia's former ambassador to Myanmar, said international expectations should be "realistic" in terms of what the visit might achieve, and that Myanmar's rulers had their own reasons for asking Ban to come.
"It is quite a political triumph to get him to go there – whether that means they're prepared to make any kind of concessions to him is something that we'll have to wait and see," he said.
"They have their own political agenda which is very rigid."
Human rights groups have expressed caution over Ban's visit, warning that it could be manipulated by the military government and used to give legitimacy to Aung San Suu Kyi's trial.
The charges against the Nobel peace laureate centre on an uninvited visit to her home by an American man, John Yettaw, who sneaked past security to swim to her lakeside home and stay for two days.
She has pleaded not guilty to the charges, but faces up to five years in jail if convicted.
Critics have condemned the trial - held behind closed doors in Yangon's Insein jail - as a sham designed to keep Aung San Suu Kyi behind bars during elections the military has scheduled for next year.
In a statement, US-based Human Rights Watch urged Ban to make the trip "meaningful" after years of failed UN attempts to win the opposition leader's freedom and promote democratic reforms.
"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 Kenneth Roth, the executive director of Human Rights Watch.
"He should make it clear that the time for stalling and playing games is over and that real change is needed now."