Save Sekhmeka Group Egypt said the statue of a court official was a symbol of its ancient civilisation, and "belongs to the Egyptian people".
The group wants the Egyptian government to step in and claim the statue.
Spencer Compton, the second Marquis of Northampton, acquired the figure during a trip to Egypt in 1850.
The statue was presented to Northampton museum by his son some years later.
The statue was sold by Christie's of London in July last year to an unknown buyer. The Department for Culture, Media and Sport (DCMS) then imposed an export bar which expires on 29 July.
A DCMS spokesman said: "The temporary export bar may be extended to 29 March 2016 if a serious intention to raise funds to purchase the statue is made.
"Anyone interested should contact the Arts Council before the current deferral period expires."
Northampton Borough Council said any challenge to the export was a matter between the government department and the buyer as it had received the money for the sale.
The group in Egypt said: "Selling a statue owned by the Egyptian people is a crime against all international norms and standards.
"The British government need to act fast and abide by international law. We call on... David Cameron to intervene to reserve (retain) Sekhemka in the UK and to extend the deadline of release so that the Egyptian government can recover the statue."
Christie's said that any challenge to the sale was "a matter between the owner and the export licence committee".
A campaign group in Egypt has joined the fight over the £15.76m sale of a 4,000-year-old statue by Northampton Borough Council.