Two US rock climbers, one of whom once pushed an Islamist terrorist over a cliff, are expected to finish climbing one of the hardest and steepest walls in the world within the coming days

For the past 2½ weeks, Kevin Jorgeson, 30, of California, and Tommy Caldwell, 36, of Colorado have been slowly making their ascent up 900 metres of granite as they climb the Dawn Wall of El Capitan in California's Yosemite National Park.

What makes their climb unique is that, if successful, they will be the first climbers to "free climb" the rock face described as "smooth as alabaster, as steep as the bedroom wall". Free climbing involves climbing the rock using just one's body, without any artificial aids, and ropes used only to protect falls.

But the expedition is just one more hurdle in Caldwell's eventful life.

Caldwell managed to become one of the world's best climbers after losing his left index finger while renovating a table 14 years ago. Doctors said he could have kept his finger, but it would never be strong enough to grip a rock so he decided to have it removed.

In 2000, Caldwell and three other climbers were kidnapped by Islamist terrorists while rock climbing in Kyrgyzstan in central Asia. They escaped after Caldwell pushed one of his captors over a cliff.

The terrorist was part of the Islamic Movement of Uzbekistan, which was fighting against the Kyrgyz government

The rebel survived the fall, but was later captured and sentenced to death by Kyrgyz soldiers.

Caldwell told National Geographic: "I couldn't believe I'd just done that, because it's something that I never morally thought I could do and I never wanted to do."

On Tuesday afternoon (AEST), Caldwell uploaded a photo to Instagram and seemed upbeat about finishing the climb as soon as possible.

"Tomorrow we leave our bacecamp [sic] and blast towards the top," Caldwell wrote on his Instagram account.

"With some luck we will be standing on top in a couple days. I an [sic] excited to walk on flat ground again although I am sure I will truly miss this experience."

Climbing Magazine writer and digital media specialist Kevin Corrigan, who has been covering the climb, said a conclusion was not too far away.

"While anything can still happen, a surprise storm, rock fall, or injury could jeopardise the effort, success is beginning to feel imminent," Corrigan wrote.

"They're expected to reach the summit tomorrow evening [Thursday morning AEST]."

There are 100 routes up to the top, with Jorgeson and Caldwell opting to take the toughest option called Dawn Wall, which looks east and faces the morning sun. They have been climbing the cliff face in 32 sections.

On Monday evening local time, Jorgeson pushed to the top of the Wino Tower; the last of the three hardest sections of the climb.

Given the tortuous nature of the cliff face, Jorgeson and Caldwell have been resting, eating and sleeping in hanging tents suspended high above the Yosemite Valley floor with spectacular 180-degree views.

They began their climb on December 27 and dozens of media outlets have rushed be there to capture the moment when they reach the top. If the pair reach the summit this week, it will mark more than 500 consecutive hours suspended from the side of the cliff.

The first successful climb of the Dawn Wall was in 1970.

"What makes the Dawn Wall so special is that it's almost not possible," renowned climber Alex Honnold told The New York Times. "The hardest pitches on the Dawn Wall are harder than I've ever climbed."

Jorgeson and Caldwell each took a mobile phone and they have been posting regular updates on social media for their family and fans.

"I am really feeling the Yosemite love these days," Caldwell wrote on his Facebook page on Sunday. "After a lifetime of climbing here I am still struck by the grandness and beauty of this place."

To anyone writing about #dawnwall, this is not an effort to "conquer." It's about realizing a dream. — Kevin Jorgeson (@kjorgeson) January 13, 2015

Wide awake before 5am alarm. Questing off Wino Tower today to the Ship's Bow. Will bivy there, 4 pitches from summit. pic.twitter.com/qTy1pX2cbi — Kevin Jorgeson (@kjorgeson) January 13, 2015