Two families who believe their babies may have been killed by Lucy Letby want the police to investigate, The Telegraph can reveal.

Parents of infants treated at the Countess of Chester Hospital, where Letby murdered seven babies and attempted to kill six others, have raised concerns about the way their children died.

Both mothers could be considered vulnerable – one is from Lithuania and does not speak much English, while the other has learning difficulties and struggles with reading and writing. Both believe Letby was present when they were at the hospital with their children.

The nurse appeared to have signed one baby’s baptism book, while the other infant’s father recognised Letby from media coverage of the trial.

Police officers contacted the families during their initial investigation but they were not included in the original case, which concluded earlier this month.

Detectives are now working through the admissions of 4,000 babies to neonatal units at the two hospitals where Letby worked and are looking for any suspicious incidents in which she may have been involved.

Emily Morris, whose one-month-old son Alvin died in tthe Countess of Chester Hospital neonatal unit after he was christened in 2013, told The Telegraph: “We don’t believe we’ve had justice.”

On Friday, Rishi Sunak said he wanted the inquiry into Letby’s crimes to be led by a judge to ensure there is a “strong, independent voice” who can “get to the bottom of what happened”.

The Prime Minister said: “Obviously this was one of the most despicable, horrific crimes in our history and it’s really important that we get answers, particularly for the families of the victims.”

Downing Street said the Government had not yet decided whether the inquiry should be put on a statutory footing, meaning it would have the power to compel witnesses – including hospital bosses – to attend.

It comes amid growing anger over the alleged failure of hospital managers to take action after concerns were raised about the killer nurse. Amanda Pritchard, the chief executive of the NHS, will hold an urgent meeting next week to discuss potential new powers to discipline hospital managers for “serious misconduct”.

On Friday, it emerged that Tony Chambers, the Countess of Chester Hospital’s former chief executive – who has been accused of shutting down concerns about Letby – was paid up to £80,000 by the trust for nine months after he resigned.

It is understood that, for six months of this period, Mr Chambers was seconded to another trust and spent a further three months preparing for the secondment.

Letby will die in jail after being sentenced to a whole life order by Mr Justice Goss for her “campaign of child murder” that showed a “deep malevolence bordering on sadism”.

Addressing the 33-year-old at Manchester Crown Court on Monday, he said: “The babies you harmed were born prematurely and some were at risk of not surviving, but in each case you deliberately harmed them, intending to kill them.”