Beblawi, a 76-year-old former finance minister, said he was prepared to appoint Brotherhood ministers as a first step towards healing Egypt's political divide.
But that olive branch was rejected by the Brotherhood.
"We do not deal with putschists. We reject all that comes from this coup," Tareq al-Morsi, a spokesman for the movement, said.
The Brotherhood has called for an uprising to restore the government of Morsi, who was ousted and detained on July 3 after a year in office following mass protests accusing him of authoritarianism and incompetence.
Thousands of his supporters gathered late last week outside the Republican Guard facility in Cairo's Nasr City district, believing he was held there.
The Brotherhood claims 51 of them, including women and children, were shot dead in the early hours of Monday morning after soldiers and police opened fire while they were praying.
The army says it was responding after coming under attack from "terrorists", whom it claims tried to storm the Republican Guards club. It says several members of the security forces were killed in the incident.
The authorities' version of events came under fire on Wednesday from human rights groups, including Amnesty International, which called for an "independent and impartial" inquiry after accusing the security forces of using "intentional lethal force".
"Despite claims by the military that protesters attacked first during clashes on Monday and that no women and children were injured, first-hand accounts collected by Amnesty International paint a very different picture," said Hassiba Hadj Sahraoui, the group's deputy regional director.
"Even if some protesters used violence, the response was disproportionate and led to the loss of life and injury among peaceful protesters.
"As politicians squabble over who started the violence, it's clear that unless the security forces are reined in and clear orders given on the use of force, we're looking at a recipe for disaster."
Fifteen Egyptian human rights groups also issued "strong condemnation" in a joint statement.
Some 88 people have been killed and 1500 wounded since last Friday, Amnesty says.
Fears of further violence were fuelled after an attack by Islamist militants on a checkpoint in the lawless Sinai peninsula left two people dead and another six injured.
In a fillip to Egypt's new rulers, Kuwait pledged £2.7 billion ($5 billion) in cash, loans and fuel on Wednesday to aid the country's collapsing economy. The promise came a day after Saudi Arabia and the United Arab Emirates promised a total of £5.4 billion.