Joseph, who was named interim prime minister in April after the resignation of Joseph Jouthe, has taken the reins of power so far, spearheading the government's response to the assassination, appealing to the United States for support and declaring a 15-day state of emergency.
Elections minister Mathias Pierre said Joseph would keep that role until presidential and legislative elections are held on 26 September.
Henry said the new government he was forming, however, would instead create a new electoral council–given the previous one had been considered too partisan–and the council would determine new dates for elections.
"I don't know what date they will be, all I know is that we cannot take too long to do these elections. We need to do them as soon as possible," Henry said.
Haiti's Senate, which currently comprises just a third of its usual 30 senators, nominated its head Joseph Lambert on Friday to be interim president, a document reviewed by Reuters showed.
Henry said he would be "ready to lead the country with all those who wanted to move forward" when asked if he was willing to rule alongside Lambert.
Haiti's 1987 constitution stipulates the head of the Supreme Court should take over as interim president. But amendments that are not unanimously recognized state that it be the prime minister, or, in the last year of a president's mandate–as was the case with Moise–that parliament should elect a president.
Further complicating the situation, the head of the Supreme Court died last month after contracting Covid-19 amid a surge in infections in one of the few countries yet to start a vaccination campaign.
There also is no sitting parliament as legislative elections scheduled for late 2019 were postponed amid political unrest.
Haitian authorities have said that Moise was killed by a group of foreign, trained assassins comprised of 26 Colombians and two Haitian Americans.
Seventeen of the men were captured after a gun battle with Haitian authorities in Petionville, a hillside suburb of Port-au-Prince, while three were killed and eight are still at large, according to Haitian police.
Colombian investigators took a Friday evening flight to Port-au-Prince to help Haitian authorities with the probe into the assassination, general Jorge Luis Vargas, head of Colombia's national police, said on Twitter.
"Traveling are the head of the National Intelligence Directive, and on behalf of the Police, the director of intelligence brigadier general Norberto Mujica and a most high-level official from INTERPOL Colombia," Vargas tweeted.
The United States has no plans currently to provide military assistance to Haiti, according to a senior administration official in Washington, after Port-au-Prince made a request for US troops on Friday.