Two Americans trying to become the first to scale a sheer 900-metre face of the El Capitan rock formation in California's Yosemite National Park without bolts or other climbing tools have both cleared a key stretch and believe they will make it to the top by Wednesday.

Slowly but surely, Tommy Caldwell and Kevin Jorgeson have been working their way up 32 "pitches", varying lengths of rock, until eventually they have climbed the whole wall.

They use the safety ropes only in case of falls, and ropes and other tools to move back and forth from their campsite perched high on the rock. Otherwise it is all hands, feet, fingers and toes.

Caldwell, 36, of Colorado, completed pitches 19 and 20 of the near-vertical granite face of El Capitan's so-called Dawn Wall using only his hands and feet late last week, spokeswoman Jess Clayton said.

His climbing partner, Kevin Jorgeson, 30, of Santa Rosa, California, finally caught up over the weekend. He had been behind Caldwell for about a week as they try to scale El Capitan's Dawn Wall without climbing aids other than safety ropes.

Jorgeson joined Caldwell at a rare ledge after his third attempt to inch his way up pitch 15, according to a spokeswoman for the attempt, Jess Clayton.

Jorgesen celebrates with a cheer after finally conquering pitch 15 on the sheer wall of El Capitan. -- PHOTO: FACEBOOK

From there, the pair is now making the final push to the summit.

Into their third week of climbing, the pair are two-thirds of the way up the wall at a spot known as Wino Tower. It was where Warren Harding, leader of the first team to climb El Capitan, paused during his ascent in 1970, using climbing tools.

"This section was the last big question mark in Tommy's mind and mentally represented a huge turning point for him," Reuters reported Clayton saying when Caldwell became the first of the pair to get past the crucial spot. "He now has full confidence that he can make it to the top. The remaining 300 metres of climbing, while still intimidating and challenging, are technically easier."

Caldwell had "tears of elation" in his eyes when he reached Wino Tower after a difficult climb through pitch 20, she said.

"There are tiny little holds, but they're far apart and facing different directions," Josh Lowell, a filmmaker who has been recording the pair's attempts for several years, told the New York Times. "Some you grab in awkward ways, sometimes barely by your fingertips, sometimes cocking your wrist in weird angles."

The route goes that way because the rock is too smooth above and below it, he said, and the stripe has just enough of a sandpaper texture to cling to.

Jorgeson struggled for more than a week on pitch 15, writing on Instagram about how razor sharp rock ledges had ripped the skin off his fingers.

On his Facebook page, he said he had run into "some of the smallest and sharpest holds I have ever attempted to hold onto. It's crazy to think that the skin on our fingertips could be the limiting fact towards success or failure".

Jorgeson showed off his bloodied hands from his attempt to climb pitch 15 on his Facebook page. -- PHOTO: FACEBOOK

Jorgeson rested two days to regrow enough skin on his fingers to try again, she said, before finally succeeding.

"Momentum is a powerful force," Jorgeson wrote in a post on his Facebook page "When it's on your side, everything feels a bit easier. When it's not on your side, it feels like wading through mud. For 7 days, my momentum was halted by Pitch 15. It took everything in my power to stay positive and resolved that I would succeed."

A growing crowd at the foot of the Dawn Wall cheered and cried when Jorgeson finally grasped a razor sharp hold that had eluded him for days, clipped an anchor to secure himself and then shouted - partly in triumph and partly in relief, according to online accounts by colleagues at the scene reported by Britain's Daily Mail.

"Pure joy," he wrote after the triumphant moment on his Facebook page. He also posted a photograph showing his bloodied hands wrapped in bandages.

Because the warmth of the day can cause their hands and feet to perspire, Jorgeson and Caldwell often start climbing as night starts to fall.

The duo first tried to scale the rock face in 2010 and 2011 before returning for this attempt on Dec 27.