If all goes as planned, 30-year-old Kevin Jorgeson of Santa Rosa, California, and 36-year-old Tommy Caldwell of Estes Park, Colorado, should reach the summit of El Capitan early Wednesday afternoon.
VIDEO: Wife of El Capitan climber awaits husband's ascent to top
The world has been watching the pair's grueling half-mile journey up the peak's Dawn Wall route. But spokeswoman Jess Clayton says the men won't give media interviews at the top. They plan to discuss the climb Thursday.
For 18 days, the two men have been attempting what many thought impossible. They are free-climbing to the 3,000-foot (914-meter) summit, meaning they don't use climbing aids, just harnesses and ropes to prevent deadly falls.
VIDEO: Father of El Capitan climber says son has always climbed
Each trained for more than five years, and they have battled bloodied fingers and unseasonably warm weather.
Jorgeson fell 11 times over seven days trying to get past one tough section. He took to Facebook on Sunday to publicly celebrate his victory.
"It took everything in my power to stay positive and resolved that I would succeed," he wrote of his continued attempt to get past the grueling section.
VIDEO: Fellow climber attests to difficulty of El Capitan climb
The climb began on Dec. 27 and was expected to take two weeks. If the men finish Wednesday, they will have been on the wall closer to three weeks.
El Capitan, the largest granite monolith in the world, has about 100 routes to the top. The first climber reached the summit in 1958.
In 1970, Warren Harding and Dean Caldwell - no relation to Tommy Caldwell - climbed Dawn Wall using ropes and countless rivets over 27 days. The duo turned down a rescue attempt by park rangers in a storm.
Photos from Caldwell's Instagram page show their climb.