The two discussed the matter during a break at a high-level meeting on Monday at the United Nations headquarters in New York, says Foreign Secretary Md Shahidul Haque.
"The US president said 'We stand by you on the Myanmar issue. And we will see how it can be resolved'," he said.
The 72nd General Assembly is the first time world leaders have gathered for a session since Trump took office. The ‘High-Level Meeting on UN Reform’ was organised by the US president.
The UN meeting comes amid a military crackdown in Rakhine state that has caused hundreds of thousands of Rohingyas to flee across the Myanmar border to Bangladesh.
Bangladesh, which has housed almost half a million Rohingya refugees for decades, has seen a new influx of over 400,000 more since the operation began in late August.
UN agencies fear the number of new refugees may reach a million by the year’s end if the situation in Rakhine remains unchanged.
Foreign Secretary Haque said Trump also asked Hasina about how the Bangladeshi economy was doing, to which the prime minister responded that 'things were going well'.
The Rohingya issue was also discussed when Indian Foreign Minister Sushma Swaraj met the prime minister.
"Sushma Swaraj said that India always stands by Bangladesh and will help to resolve the crisis," said Haque.
Hasina and Swaraj took the same Etihad Airways flight to New York from Abu Dhabi, he said, adding that they had a 'brief meeting' on board.
The Bangladesh prime minister is scheduled to address the UN General Assembly on Sept 21, when she will present her country's proposals on solving the refugee crisis.
Hasina is also scheduled for a bilateral meeting with UN Secretary-General Antonio Guterres the same day.
On Monday, the Bangladesh prime minister met with the chief of the UN High Commissioner for Refugees Filippo Grandi.
"The UNHCR wants to help Bangladesh on the issue of those who fled Myanmar," said Bangladesh Foreign Secretary Haque.
The same day, the British foreign secretary organised a session on the Rohingya crisis, which the Bangladesh foreign minister and Myanmar's national security adviser attended along with representatives from several nations, including China, Russia, Germany, the US, Sweden, Switzerland and Denmark.
"The issue has been discussed extensively there. Bangladesh has been praised for providing shelter to refugees while Myanmar has been urged to end the violence and take back its people," said Haque.
He said the Myanmar national security adviser tried to raise the issue of whether the international media were reporting the true facts of the situation in Rakhine state.
"He was told that everyone was aware of what is really going on there and that the session will focus on how to solve the crisis, rather than discussing that," said the Bangladesh foreign secretary.
Prime Minister Hasina also held a meeting with her Bhutanese counterpart Tshering Tobgay.
They discussed whether Bhutan can use the satellite Bangladesh will launching into orbit and buy bandwidth from the second submarine cable, according to Secretary Haque.
Hasina also attended two more sessions on Monday – the 'High-level Meeting on the Prevention of Sexual Exploitation and Abuse' and the 'High-level Follow-up Meeting on Global Deal for Decent Work and Inclusive Growth.'