A prosecutor is seeking charges against Henry, who has been asked to explain his links with a key suspect in the killing, Joseph Felix Badio.
Records show the two men had multiple phone calls just hours after the assassination, prosecutors say. Moïse was killed at home on 7 July.
He was shot dead after gunmen stormed his private residence in a suburb of the capital, Port-au-Prince. His death triggered a political crisis which was exacerbated when a huge earthquake struck the south of Haiti a month ago.
On Monday, Henry sent a letter purportedly sacking Chief Prosecutor Bed-Ford Claude and accusing him of a “serious administrative offence”. He later nominated a replacement.
However, on Tuesday Claude appeared to remain in his post as he asked a judge investigating the murder of Moïse to charge the prime minister over his “suspected” involvement in the case.
Sources with knowledge of Haiti say it is not in the prime minister’s remit to dismiss the prosecutor.
Haiti’s justice minister has ordered extra police protection for Claude following a number of reported threats.
What’s the PM’s alleged link to the case?
Prosecutors say records obtained from phone operator Digicel have confirmed that Mr Badio – a justice ministry official accused of planning the assassination – and Henry spoke twice shortly after Moïse’s killing.
Geolocation data also showed that Badio, who disappeared after the murder and is currently being sought by police, was speaking from somewhere close to the scene of the crime. He has not publicly commented on these claims.
The prime minister recently rejected what he called “diversionary tactics” intended to “sow confusion” without responding to the specific accusations against him.
Henry has previously told local media that he knew Badio. He has defended him and said that he did not believe he was involved because he did not have the means.
The prime minister was reportedly given a deadline of Tuesday 14 September to supply the investigation with information.
It comes as authorities step up their efforts to arrest additional suspects in the assassination.
Police say there are now 44 people held in custody in connection with the plot, including 18 retired members of the Colombian military.
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