Buckingham Palace staff may also face questions over what they knew and what they did with that information.
Ms Giuffre took her own life in April.
Ed Miliband, the Energy Secretary, said developments were “deeply concerning”.
Asked about the reports on the BBC’s Sunday With Laura Kuenssberg, he said: “These are deeply concerning allegations. I think people want to look at those allegations and what the substance is behind them.
“But if that is correct, that is absolutely not the way that close protection officers should be used.”
The revelations, first published by the Mail on Sunday, came just 24 hours after the King moved to act against Prince Andrew amid fears that a newly unearthed email sent to Jeffrey Epstein would open the floodgates to more scandal, The Telegraph understands.
The Prince was forced to give up his remaining royal titles after it emerged he had stayed in touch with the convicted paedophile longer than he previously admitted.
The email exposed a fundamental “fault line” in Andrew’s defence and proved a “tipping point” for the King, palace sources said.
The matter was then dealt with swiftly, with Andrew pressed to accept the latest sanction in a matter of hours. His ex-wife, Sarah Ferguson, has also given up her title.
To formally strip the Prince of his titles, Buckingham Palace would have had to enlist the help of Parliament. But sources said they did not want to waste parliamentary time with what had become a “family distraction”.
The matter could yet be taken out of their hands, however, if there is public clamour for MPs to act.
‘ The preceding article may include information circulated by third parties ’
‘ Some details of this article were extracted from the following source www.telegraph.co.uk ’
