Patel also faces the prospect of an inquiry following Rutnam's decision to sue the government over his alleged forced exit and Starmer called for the head of the UK civil service, Cabinet Secretary Mark Sedwill, to start "an immediate investigation" into the circumstances surrounding his departure.
"There are now urgent questions that must be answered and steps that need to be taken," he said.
John McDonnell, the shadow chancellor, described the situation as "unprecedented", which put a question mark over Patel's longevity in the Home Office as minister.
Labour's Yvette Cooper, the chair of the home affairs select committee which holds the Home Office to account, said the row reflected extremely badly on the government".
She said: "To end up with one of the most senior public servants in the country taking court action against one of the great offices of state shows a shocking level of breakdown in the normal functioning of government.
"For the home secretary and prime minister to have allowed things to reach this point is appalling, especially at a time when the Home Office faces crucial challenges with rising violent crime, forthcoming counter-terror legislation, new immigration laws, and sensitive negotiations on post-Brexit security cooperation."
Dave Penman, general secretary of the FDA - the trade union for senior civil servants in the UK, which will back Rutnam's legal case said it is expected that Rutnam will demand a public hearing at which he can outline his case against Patel.
"The Home Office now needs to find new leadership at a time when it needs stability. Those who engage in anonymous briefings need to bear the responsibility for this destructive behaviour," he said, in reference to some unattributed media reports over the past month around clashes between the minister and her top civil servant.
On Saturday, Rutnam announced his resignation in an emotional television statement as he claimed a vicious and orchestrated campaign against him and pointed the finger of blame at Patel.
The Home Secretary [Patel] categorically denied any involvement in this campaign to the Cabinet Office. I regret I do not believe her, said Rutnam.
Even despite this campaign, I was willing to effect a reconciliation with the Home Secretary, as requested by the Cabinet Secretary on behalf of the Prime Minister. But despite my efforts to engage with her, Priti Patel has made no effort to engage with me to discuss this, he said, adding that he had very strong grounds to claim constructive, unfair dismissal, and I will be pursuing that claim in the courts.
Rutnam also referred to the tensions with Patel when he encouraged her to change her behaviour.
"I have received allegations that her conduct has included shouting and swearing, belittling people, making unreasonable and repeated demands, behaviour that created fear and needed some bravery to call out, he claimed.
The UK Home Office or Patel herself are yet to officially comment on the resignation.
Patel, who joined the Boris Johnson-led Cabinet in July 2019, has previously been forced to resign as Secretary of State for International Development by former Prime Minister Theresa May in 2017 over a controversy surrounding some unofficial meetings with Israeli ministers and business executives during a visit to Israel.
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