Swaraj met Hasina at Hasina's office at 11.30 a.m. Thursday.
"Swaraj handed over the letter during a meeting at the prime minister's office," Xinhua quoted Bangladeshi Foreign Minister Mahmood Ali as telling reporters.
Swaraj called on Hasina to discuss key bilateral issues, sources said.
According to the sources, they discussed trade and investment, security, connectivity, and border management, among other issues.
Bilateral issues between the two countries came up for discussion during the meeting, which lasted for about one hour.
Earlier in the day, Swaraj met Bangladeshi Foreign Minister Abul Hassan Mahmood Ali at 10 a.m. for official talks and discussed bilateral issues.
"India, Bangladesh begin discussion on entire gamut of bilateral relations," India's external affairs ministry spokesman Syed Akbaruddin tweeted.
Swaraj is also scheduled to meet Bangladesh President Abdul Hamid at Bangabhaban at 6 p.m. Thursday.
This is her first official visit abroad as the external affairs minister.
Swaraj will also meet Bangladesh's Leader of Opposition Raushan Ershad, BNP chairperson Khaleda Zia and prime minister's advisers Gowher Rizvi and Mashiur Rahman before leaving for India Friday.