It comes after her counsel accused the home secretary of not engaging “more constructively” with the process.
The Royal Courts of Justice heard how coroner Baroness Heather Hallett previously requested Ms Patel’s view on whether the inquest into Dawn Sturgess' death three years ago could be widened to a public inquiry.
Ms Patel declined, saying it would be “inappropriate” to do so in advance of a formal request.
And so, Lady Hallett said she would write to the Home Office to formally request a public inquiry. A response to the request is expected by the end of this year.
Any public inquiry is unlikely to start before 2023, and will be held partly in private for security reasons.
What happened in Salisbury and Amesbury in 2018?
Former Russian double agent Sergei Skripal, 66, and his daughter Yulia, 33, were found unconscious from the effects of the military nerve agent Novichok on a bench in the cathedral city of Salisbury on March 4.
They were admitted to Salisbury District Hospital and both survived.
Detective Sergeant Bailey was also admitted to hospital and survived.
Three months later, on June 30, Dawn Sturgess and her partner Charlie Rowley fell ill at their home in Amesbury, near Salisbury.
Ms Sturgess, 44, died in hospital eight days later having never regained consciousness.
They were exposed to the nerve agent from a perfume bottle discarded by those responsible for the attack on Mr Skripal and his daughter.
Announcing her decision, coroner Lady Hallett said: “As set out in my letter to the Secretary of State for the Home Department, I have firmly concluded that I cannot conduct a full, fair and effective inquest into the death of Dawn Sturgess if these proceedings continue as an inquest.
“I have seen substantial material that is far too sensitive to be made public, even in gisted form.
“I have no option, therefore, but to request the Secretary of State convert this inquest into an inquiry and I invite her to consider my request as a matter of urgency.”
She added: “I do understand the difficulties in government… but I really do hope the Secretary of State can respond within the timescale considered.”
'What difference is it going to make? None whatsoever': Charlie Rowley says the news that a third suspect has been charged has given him no fresh hope for justice
Cathryn McGahey, a lawyer representing the government, said Ms Patel would “very much aim to make a decision” by Christmas.
The coroner said she would request Ms Patel’s reply within two months.
Lady Hallett’s counsel Andrew O’Connor outlining the correspondence between the coroner and the home secretary at Wednesday’s pre-inquest hearing and said: “It is a matter of considerable regret that the Secretary of State (Ms Patel) has not engaged more constructively with the coroner’s request.
“Since you have now reached the clear view as to the need for an inquiry, it would have assisted all involved to know the Home Secretary’s provisional view on this issue.”
Mr O’Connor said the Home Office seemed to consider Lady Hallett’s request for Ms Patel to read the case documents “too burdensome".
Adam Straw, representing Ms Sturgess’ family and her partner Mr Rowley, supported a public inquiry.
He said: “Their overriding concern is to ensure the truth of how Ms Sturgess died is established.
“No family should wait five-and-a-half years to find out how someone died.”
His colleague, Michael Mansfield, said the prospect of a criminal trial was “infinitesimal”, and so a public inquiry represented the best way of finding out what happened.
If granted, the inquiry would likely be held in Salisbury Guildhall and at venues in London.
A further pre-inquest review into Ms Sturgess' death will be held in December.
On Tuesday, a third Russian spy was charged with attempted murder over the Salisbury poisonings in 2018.
Denis Sergeev, who operated under the cover persona of Sergey Vyacheslavovich Fedotov, is thought to have arrived in at Heathrow Airport four hours before the other two suspects - Alexander Petrov, 41, and Ruslan Boshirov, 43, arrived at Gatwick Airport on March 2, 2018.
He met up with the pair in London but did not go to Salisbury, where the first poisoning incident happened.
Petrov and Boshirov previously denied any involvement in the plot and claimed they were only in Salisbury for a sightseeing tour of the cathedral.