Letby, 35, is serving life in jail after being found guilty of murdering seven newborns and attempting to murder eight more between June 2015 and June 2016 while working at the Countess of Chester Hospital in northern England.
Letby, Britain's worst serial child killer of modern times, has maintained her innocence throughout but has been refused permission to appeal against her convictions.
She has been told she would never be released.
Following her jailing, police began to investigate possible corporate manslaughter offences at the hospital, an inquiry that was later widened to consider if individuals might be guilty of gross negligence manslaughter.
Detective Superintendent Paul Hughes, who is leading the investigation, said three unnamed members of the hospital's senior leadership team had been arrested on Monday.
They were subsequently released on bail, pending further inquires.
"Both the corporate manslaughter and gross negligence manslaughter elements of the investigation are continuing and there are no set time scales for these," Hughes said in a statement on Tuesday, adding this did not affect Letby's convictions.
Hughes said a police investigation was continuing into whether Letby had committed more crimes at the hospital and at another unit where she had previously worked.
Since she was jailed, some MPs, medical experts and other specialists have publicly challenged the prosecution evidence that was used to find her guilty.
The Criminal Cases Review Commission, which examines potential miscarriages of justice, is considering an application from Letby's legal team, and last month, former health minister Jeremy Hunt said the case should be urgently re-examined.
Letby's lawyer Mark McDonald said the police announcement came at a sensitive time.
"Despite this, the concerns many have raised will not go away, and we will continue to publicly discuss them," McDonald said in a statement.
Kate Blackwell, a lawyer for the hospital's senior managers, told a public inquiry examining how the murders were able to take place that there were other explanations for the deaths which needed to be considered.
The managers strongly denied accusations they "deliberately and knowingly" protected a murderer, Blackwell said.
Police have previously said scepticism about the case was based on partial knowledge of the facts presented to the courts.
The findings of the public inquiry are due to be published early next year after the chairman rejected calls for her work to be paused.
