Chevron, one of the biggest foreign investors here, supplies half Bangladesh's natural gas needs, while ConocoPhillips is exploring for gas in the deep waters of the Bay of Bengal.
Delwar Hossain, head of the Department of International Relations at Dhaka University, said Clinton's visit could be a milestone for Bangladesh.
"If Bangladesh can get duty free access of garment products and push for a better relationship for future security cooperation, that will be a big achievement," he said.
Clinton is also expected to raise the issue of Nobel Peace Laureate Muhammad Yunus' ouster from his Grameen Bank, which pioneered providing small loans to the poor. The government said the 71-year-old Yunus, a friend of the Clinton family, was well past the retirement age of 60 and forced him out last year.
As he fought unsuccessfully to keep his job, Yunus spoke to Clinton on the phone and later met with her in Washington to discuss the future of Bangladeshi civil society, according to the State Department.
Yunus' allies said the ouster was political and pointed to Hasina's anger at his 2007 effort to form a political party backed by the powerful army when the country was under a state of emergency and Hasina herself was behind bars.
Bangladesh is also seeking the repatriation of Rashed Chowdhury, who is facing a death sentence for his role in the 1975 assassination of independence leader Sheikh Mujibur Rahman, Hasina's father, during a military coup. The government says he lives in the United States, and the two nations have no extradition treaty.
As Bangladesh modernizes its military, the U.S. also is trying to persuade it to buy military surplus from Washington instead of weapons from China and Russia.