Government spokesman Mohammed Momani told the Associated Press "the number is 12,000, but it is increasing".
Last month, he said a similar estimate by the UN was "greatly exaggerated".
Mr Momani said about 50 to 100 refugees were being allowed into Jordan each day, with priority given to women, children, and elderly and sick people.
Mass entry was unlikely because of security concerns, he added.
On 8 December, the UN High Commissioner for Refugees (UNHCR) said the number of Syrians stranded in deteriorating humanitarian conditions, beside an earthen berm on the Syria-Jordan border, had tripled.
About 11,000 people were massed at Rukban, about 8km (5 miles) to the west of the point at which the Iraq, Syria, and Jordan borders meet, and 1,000 at Hadalat, some 90km further west, spokeswoman Melissa Fleming told reporters.
They included children, women and others who were "vulnerable and really need help", she added.
Many of them are believed to have fled air strikes by Russia, which supports Syrian President Bashar al-Assad, on areas controlled by the jihadist group Islamic State (IS) in eastern Homs province and neighbouring Raqqa.
Ms Fleming warned that if the refugees were not admitted and substantial assistance was not provided, their lives would "be at risk in the coming months".
She said that Jordan had legitimate security concerns, but that they could be addressed by properly assessing each case.
Jordan is hosting 633,000 of the 4.39 million Syrians registered as refugees with the UN. The government says more than one million other Syrians are living there, including those who arrived before the uprising against Mr Assad erupted in 2011.
Jordan has acknowledged that thousands of Syrian refugees are stranded in a remote desert area on its northern border with Syria awaiting entry.