Helmand has seen the worst violence anywhere in Afghanistan, and military commanders say they need to break what they call the stalemate in the south of the country, says the BBC's Martin Patience in the Afghan capital, Kabul.
The captured soldier was not involved in the operation, codenamed Khanjar, or Strike of the Sword.
A hardline Taliban faction called Haqqani said it had the soldier, but this has not been confirmed by the main Taliban spokesman.
The army was using all its resources to find the missing serviceman, who was taken on Tuesday, spokeswoman Capt Elizabeth Mathias said.
AFP news agency said a commander of Haqqani, named only as Bahram, said the soldier was captured along with three Afghans in the Yousuf Khail district of Paktika province.
The commander said the soldier had been taken to "a safe place".
Another Haqqani commander, Mullah Sangeen, told Reuters the soldier would be held until Taliban fighters detained by the US were released.
Caroline Wyatt BBC defence correspondent
The US marines say the operation will be decisive and is intended to seize almost the entire lower Helmand River valley, which remains the heartland of the Taliban insurgency in Helmand and a major area for the production of opium, which helps fund the insurgency.
US commanders hope this offensive will help turn the tide in the current stalemate against the Taliban. The aim of this major operation is to "clear, hold and build" in areas under Taliban influence ahead of Afghan presidential elections this August.
The hope is also that if US and other Nato forces are seen as clearly winning the military battle against the insurgents, middle and lower-ranking Taliban leaders and fighters are more likely to defect back to the Afghan government's side, with US and UK sources in Kabul convinced that the US troop surge is already unsettling the Taliban leadership.
Thousands of British forces under Nato command have been fighting the Taliban in Helmand since 2006, but there has been criticism that they have been overstretched and under-resourced.
One of two British soldiers killed in an explosion in Helmand province on Wednesday was the highest-ranking Army officer to die since the Falklands war of 1982, the Ministry of Defence said.
He was named at Lt Col Rupert Thorneloe, 39, the commanding officer of the 1st Battalion Welsh Guards. Trooper Joshua Hammond, 18, also died, and six others injured.
General Sir Richard Dannatt, Chief of the General Staff, said Lt Col Thornloe was "an outstanding commanding officer" and his death was a "devastating blow".
He added: "At the leading edge of his generation, his loss will be felt deeply not only by his family but also by his soldiers and others, who like me, had the privilege to serve with him."
The two men were killed when a roadside bomb exploded under their Viking armoured vehicle. Lt Col Thornloe had joined a supply convoy to see his men deployed on operation Panther's Claw, to oust the Taliban from the area around Lashkar Gah.
The BBC's defence correspondent Caroline Wyatt says questions will be asked about why such a high-ranking officer was travelling a Viking vehicle.
They are supposed to be restricted to lower-risk areas, and are due to be replaced in Afghanistan next year by the more heavily armoured 'Warthog' vehicle.
About 4,000 US and 650 Afghan troops deployed to Helmand river valley
Initial operations focused on villages of Nawa and Garmsir near the provincial capital Lashkar Gah
British operation recently recaptured the village of Babaji from the Taliban to the north of the region
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