Green was forced to resign as First Secretary of State and effective deputy Prime Minister last night after a parliamentary investigation found he had breached the ministerial code of conduct.
During a tour of Poland today, Theresa May said she wanted an inquiry into the leak of information about a Metropolitan Police inquiry dating back nine years by a former officer.
"I share the concerns about the comments from a former Met officer and those concerns must be taken seriously and properly looked at," she said.
"But I shared the concerns raised when your [Green's] Parliamentary office was raided when you were a shadow home office minister. And I share the concerns raised at comments made by a former officer involved in that case," she wrote.
The "former officer" is believed to be a reference to the intervention of former Met Police computer expert Neil Lewis, who spoke out in support of claims by former assistant commissioner Bob Quick concerning pornographic images found on Green's official computer when he was a shadow minister in 2008.
"These are serious allegations and we are investigating to determine whether the law has been broken and what further action is necessary including potential criminal prosecution," said Information Commissioner Elizabeth Denham.
Green stepped down after it was found that he made "inaccurate and misleading" statements about what he knew about claims that pornography had been found on a computer in his House of Commons office nearly nine years ago.
Green's departure marks the third high-profile ministerial exit from Theresa May's Cabinet within the space of two months, after former defence secretary Michael Fallon over allegations of sexual misconduct and former international development secretary Priti Patel over undisclosed meetings with Israeli officials.
Green, who campaigned for Remain in last year's European Union (EU) referendum, had been a leading voice in the UK Cabinet for a "softer" Brexit.
As a friend of May since their Oxford University days, he was believed to be her strongest ally in an increasingly turbulent Cabinet and his exit from her front bench is seen as a major blow to her personally.
Butan official report by the UK Cabinet Office found that statements he had made about being unaware of pornographic material found on his computer were "inaccurate and misleading" and as such fell short of the ministerial code.
The report also found that although there were "competing and contradictory accounts of what were private meetings" between himself and Maltby, "the investigation found Ms Maltby's account to be plausible".
In his letter, Green said he accepted statements he made about what he knew about the pornography investigation could have been "clearer", conceding his lawyers had been informed about the original discovery in 2008 and the police had raised the matter with him in a phone call in 2013.
According to Downing Street, May might decide not to appoint a new First Secretary of State but a replacement for Green's parallel role, as Cabinet Office minster, will be announced in the New Year.
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