The bilateral meeting was held in Dhaka, Bangladesh, the Foreign Affairs Ministry said on its official twitter account, Wednesday.
It said Indonesia has attempted to address the humanitarian crisis in Rakhine State comprehensively, from its upstream in Myanmar to its downstream in Bangladesh, as a destination country for refugees.
Marsudi had earlier spoken to Ali and expressed commitment to help Bangladesh in handling the refugees.
"Again, I have spoken to Bangladeshs foreign minister. I have spoken about the humanitarian aid that we can send to the Bangladesh government to handle the refugee issue," Marsudi remarked.
The minister said Bangladesh has responded positively and lauded Indonesias offer.
"On the form (of aid), we will discuss this further. I conveyed that President Joko Widodo has expressed deep concern over the situation and assigned me to seek information on what form of help we can give to Bangladesh regarding the refugee issue," Marsudi said.
Prior to her visit to Bangladesh, Marsudi had visited Myanmar to meet State Counselor and Foreign Minister Aung San Suu Kyi and several high-ranking officials on Monday to address the ongoing armed skirmishes in Rakhine State.
Indonesia had submitted a proposal to Myanmar calling to apply the 4+1 Formula for Rakhine State to restore peace and allow immediate access for offering humanitarian assistance in the conflict-hit region.
The formula includes four main elements to recover security and stability, maximize self-restraint and avoid use of force, protect all citizens of Myanmar regardless of ethnicity and religion, as well as open access for humanitarian assistance.
One of the positive elements is to implement the recommendations of the Advisory Commission on Rakhine State led by Kofi Annan, the former UN security general.
The attack on the Myanmar police on Aug 25 by the Arakan Rohingya Salvation Army had triggered a counter-offensive raid that resulted in the killings of more than 100 people, including civilians.
The incident has forced thousands of Rohingya Muslims to flee to Bangladesh to avoid Myanmars worst humanitarian conflict in the last five years.(*)