The picture set an auction record both for the artist and for any work on paper when it sold at Christie's late last year.
The buyer was believed to be the American businessman and money manager Leon Black, who is a trustee of the Metropolitan Museum of Art in New York.
Mrs Hodge accepted the reviewing committee recommendation that an export licence be deferred on the grounds that "the drawing is of outstanding aesthetic importance and of outstanding significance for the study of Raphael's work".
The committee awarded the drawing a starred rating, meaning that every possible effort should be made to keep it in the country.
The decision on whether to grant an export licence will now be deferred until May 25 and may then be extended until November 25 if a serious intention to raise public funds or interest from a UK-based private buyer emerges.
The money required is £29,161,250 (the hammer price plus the buyer's premium), but it seems unlikely that public bodies will find it easy to raise such an amount at a time when budgets are being squeezed.
It emerged last week that the Department for Culture, Media and Sport has been under pressure from the Treasury to halve the £10m allocation to the National Heritage Memorial Fund. The fund has already pledged £3.3m to the purchase of Titian's Diana and Actaeon from the Duke of Sutherland.
Raphael's Head of a Muse has been in the UK since at least 1850 and was consigned to Christie's by the heirs of the British collector Norman Colville.
The black chalk drawing was a study for a figure in the Parnassus fresco in the Vatican, considered one of the artist's masterpieces.
Professor David Ekserdjian, a member of the export licence reviewing committee, said: "The UK is rich in Raphael drawings, but this is the only auxiliary cartoon related to the Stanza della Signatura commission.
"It is important for the study of Raphael's work as a unique record of his original artistic vision."
By Alex Capon