Kevin Jorgeson, 30, of Santa Rosa, California, and Tommy Caldwell, 36, of Colorado will likely finish the climb up El Capitan's Dawn Wall on Wednesday evening, said Jess Clayton, who represents Patagonia, a sponsor of Caldwell.
VIDEO: Wife of El Capitan climber awaits husband's ascent to top
For 17 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.
For full coverage on Jorgeson and Caldwell's historic climb, click here.