Libyan police have surrounded the Swiss embassy in Tripoli after issuing a deadline for Switzerland to hand over two nationals sheltering in the building.
According to a Reuters correspondent on Monday, one of the businessmen at the centre of the diplomatic row, Rachid Hamdani, has already come out of the building and was driven away by Libyan officials.
But there was no sign of the second Swiss national, Max Goeldi.
Libyan officials say Hamdani, who has been acquitted of criminal charges, is free to leave if the Swiss comply with their demands.
But Tripoli says Goeldi must be handed over to serve a four-month prison term for violating immigration rules.
Last week Tripoli stopped issuing visas to most European citizens, a step that was linked to the row with Switzerland.
Goeldi was sentenced to four months in jail by a Libyan court for having allegedly overstayed his visa and conducting illegal business activity.
Following the Libyan government's ultimatum, Salah Zahaf, Goeldi's lawyer, told the AFP news agency: "He will leave the embassy and turn himself in voluntarily."
Zahaf said he expected Goeldi to be taken to Ain Zara prison near Tripoli, and that he had been instructed to ask the Libyan justice system for an amnesty.
Relations between Libya and Switzerland have been strained since July 2008, when Hannibal - a son of Muammar Gaddafi, the Libyan leader - was arrested in the Swiss city of Geneva after two workers complained he had mistreated them.
The row escalated when Libya swiftly detained Hamdani and Goeldi and confiscated their passports.
It deteriorated further last year when a tentative political deal between Swiss and Libyan ministers fell through.