It was revealed the Prime Minister had asked the Cabinet Office to "establish the facts" following allegations that Ms Patel has breached the ministerial code.
Conservative MPs have rallied in support of Priti Patel amid reports that an inquiry into allegations of bullying found that she had broken the rules for ministers.
Patel apologised for upset caused by her behaviour, saying in a statement: "It has never been my intention to cause upset to anyone".
The report noted Ms Patel was "unaware" of the impact of her behaviour because no feedback was given to her at the time. He is also reassured that relationships, practices and culture in the Home Office are much improved.
Nick Thomas-Symonds, the home affairs spokesman for the opposition Labour Party, said the full report should be published and the Independent Committee on Standards in Public Life should investigate Johnson and Patel.
In his report, Allan said Patel had "not consistently met the high standards required by the ministerial code of treating her civil servants with consideration and respect".
Health Secretary Matt Hancock told the BBC that she was an "excellent Home Secretary" who had "been nothing but courteous and kind in all the dealings I've had with her".
Publication of the investigation's findings has been delayed for months by Prime Minister Boris Johnson's office. It said her behaviour met the definition of bullying, though it may have been unintentional.
He added: "The high pressure and demands of the role, in the Home Office, coupled with the need for more supportive leadership from top of the department has clearly been a contributory factor".
Sir Philip Rutnam launched a blistering attack on Ms Patel when he quit his role at the end of February
"But I feel that it is right that I should now resign from my position as the Prime Minister's independent adviser on the Code".
He said there were "occasions of shouting and swearing" that had upset people, though he accepted that Patel may not have meant to have that effect.
He said: "It is hard to imagine another workplace in the United Kingdom where this behaviour would be condoned by those at the top".
British Prime Minister Boris Johnson on Friday stood by his interior minister, a hardline ally in the Brexiteer government, resisting calls for her resignation after an official finding that she may have bullied staff. The Government should be setting an example.
She has previously served as the Secretary of State for International Development (2016-17), Minister of State for Employment (2015-16), Exchequer Secretary to the Treasury (2014-15) and is also the incumbent MP for Witham.
"He seems to believe there is one rule for him and his allies and another rule for everyone else".
"I am very grateful for the hard work of thousands of civil servants who help to deliver the government's agenda", she added.
A Cabinet Office investigation was launched in March over allegations that Patel belittled colleagues and clashed with senior officials in three different departments.
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