Al-Hashemi has been a regular critic of al-Maliki's alleged sectarian bias, but his Kurdish and Shia colleagues vowed to hold inclusive talks in the coming days.
On Wednesday, the National Concord Front, Iraq's main Sunni political bloc, said it was withdrawing its five ministers and deputy prime minister from the government coalition, dealing a blow to the government's claims to represent all Iraqis.
After Sunday's meeting, al-Maliki said: "We agreed on making efforts in order to get the Concord Front back and to tackle what we are able to tackle, but not everything that they have demanded."
Abdel Mehdi said: "The prime minister's position is that he has refused to accept the resignations. This is a good thing and the door is open to discuss this topic."
Salim al-Jibouri, a senior member of parliament from the front, said the ministers would still quit, despite al-Maliki's decision not to accept their resignations.
He said: "We are insisting on our position. For us, the matter does not end with Maliki accepting or rejecting the resignations.
"We are talking about a programme. The issue is if he accepts or refuses to accept our programme."
The programme refers to the front's lists of demands that include disbanding Shia fighters and giving the bloc a bigger say in security matters.
Saleem Abdullah, a spokesman for the front, said: "At the moment, we have no intention of changing our position on the resignations. It is not necessary for us to be a part of the government."
However, Abdulllah also said "the door to dialogue is open, even if we are outside the government".
In a separate development, at least 13 people have been killed and 11 wounded by mortar bombs that struck a petrol station in eastern Baghdad.