The world's third-largest platinum producer said 27 percent of workers reported for the morning shift, while unions said they were in meetings to discuss the ultimatum.
The 11-day-old illegal strike by about 3,000 rock drill operators has closed production at the Marikana platinum mine, where 34 people were gunned down by police Thursday after 10 others were killed in clashes between rival unions.
As union leaders held meetings, about 1,000 workers gathered near the mine said they would not return to the job, with several accusing Lonmin of insensitivity for expecting them to go back to work while they are still in mourning.
Monday is the first day of a week of national mourning declared by President Jacob Zuma, with a memorial service planned for Thursday.
"They can fire us if they want, we are not going back to work. Zuma must shut down that mine," one worker said.
Lonmin has threatened to sack the workers who launched the strike on August 10, while urging 25,000 other employees and 10,000 contractors to return to work, assuring them they will be safe from attacks by strikers.
"The safety and security of our employees is paramount and nobody will be asked to report for duty if the police consider them in danger of reprisals," chief financial officer Simon Scott said in a statement.
The upstart Association of Mineworkers and Construction Union (AMCU), which launched the strike calling for a tripling of the basic wage of 4,000 rand ($486, 400 euros) a month, said it expected to have a decision on the ultimatum later Monday.
"We are still engaged in a meeting discussing the issue," treasurer Jimmy Gama told AFP.
The powerful National Union of Mineworkers (NUM), whose membership has been eroded by the AMCU's emergence, said it was also in a meeting.
Lonmin management was reportedly in talks with a team of cabinet ministers tasked with organising funerals and aid for victims' families.
Workers meeting in the field near the mine were gathered without weapons, in a mood far calmer than that of similar gatherings over the last week.
Few police were seen on patrol, though a helicopter sometimes passed in the distance.
A local leader addressed the crowd, reading out a list of the 259 people arrested after Thursday's crackdown, when police opened fire on hundreds of strikers armed mainly with spears, machetes and clubs.
After the dust settled, 34 were dead and 78 wounded in the bloodiest day of protest since the fall of white-minority apartheid rule in 1994.
Scores of people who were arrested were expected to appear Monday in a court outside the capital Pretoria on charges ranging from murder to public unrest. Eleven mini-buses have been hired to transport relatives to support the detainees, said the local chief addressing the crowd near the mine.
South Africa holds over 80 percent of the world's reserves of platinum, popular for jewelry but mainly used to make catalytic converters on cars, which scrub their emissions of pollutants.
Weak demand in major auto markets has taken a toll on the industry, and several mines have been forced to close this year.
Fears of layoffs have heightened tensions among mine workers, a situation the radical AMCU has exploited to gain membership with the promise of hefty wages and claims that the politically connected NUM has become too cozy in the corridors of power.