Maurer, president of the International Committee of the Red Cross, during the 45-minute meeting urged respect for international humanitarian law and the need to boost assistance on the ground in Syria, ICRC spokesperson Hisham Hassan said.
State television said Assad told Maurer that he supports the work of the ICRC in Syria so long as it remains "impartial and independent".
"President Assad assured the president of the [ICRC] that he welcomed the humanitarian operations carried out by the committee on the ground in Syria, as long as it remains impartial and independent," the television said.
Hisham described the meeting as "positive" and said Maurer had raised topics related to the "protection of civilians during hostilities".
Importance of access
These included the importance of access to "medical care and food as well as visits to detainees".
Maurer had also raised the necessity to ensure swift provision by the ICRC of humanitarian aid such as medical supplies and equipment to restore damaged water infrastructure.
Maurer arrived Monday evening in Damascus for his first visit to the war-torn country since taking over the post on July 1. Besides Assad, he has also met with Deputy Foreign Minister Faisal Mekdad and was scheduled to hold talks with other Syrian officials.
ICRC spokesperson in Damascus, Rabab Rifai said earlier on Tuesday that Maurer's mission would "focus on increased humanitarian needs and to remind the belligerents of their obligation under international law related to the protection of civilians".
Maurer, who is in Syria until Thursday, will not meet representatives of the opposition "at least under the framework of this visit," she said, adding he could make a "field trip".
Exposed to extreme violence
In a statement issued on Monday in Geneva, Maurer said: "At a time when more and more civilians are being exposed to extreme violence, it is of the utmost importance that we and the Syrian Arab Red Crescent succeed in significantly scaling up our humanitarian response."
"It is vital that we build on what has already been achieved on the ground," he added.
His talks with Syrian officials will also deal with the difficulties the ICRC and the Red Crescent face as they try to reach people affected by the armed conflict, the ICRC said.
Despite the difficult working conditions, the ICRC said it and the Syrian Arab Red Crescent had provided nearly 180 000 people with food in the hardest-hit parts of the country since mid-July.
Since the beginning of the year, the two organisations have provided relief items to more than 800 000 people and have helped more than one million people access clean water, it added. – Sapa-AFP