Mrs Clinton, the most senior American to visit the country in half a century, told the president she was “encouraged” by recent policy changes.
The US maintains tight sanctions on senior figures in Burma’s hierarchy.
But a series of reforms this year has led to speculation that decades of isolation could be about to end.
However, US officials have stressed that there is unlikely to be any major announcements on sanctions during Mrs Clinton’s trip.
Analysts say the US is more likely to consider symbolic gestures such as upgrading its mission in Burma to a full embassy.
After talks with Thein Sein in the remote capital, Nay Pyi Taw, Mrs Clinton is due to head to the main city, Rangoon, to meet pro-democracy leader Aung San Suu Kyi.
Ms Suu Kyi’s National League for Democracy recently re-registered as a political party, and she is expected to stand for parliament in forthcoming by-elections.
BBC state department correspondent Kim Ghattas, travelling with Mrs Clinton, says the top US diplomat’s visit is both a reward for the reforms that have already taken place and an incentive for Burma’s government to do more.
Mrs Clinton is the first secretary of state to visit Burma since 1955.
The country was taken over by the military in 1962 and ruled by a brutal and unpredictable junta until last year, when the army ceded power to a nominally civilian government.