The children were taken in a cross-border raid in the Gambella region last Friday, in which 208 people died.
The government has said members of the Murle community were responsible.
Flags have been flying at half mast in Ethiopia as the country mourns those who were killed.
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A government official in Gambella said that the abducted children would soon be rescued, the government-affiliated Fana Broadcasting Corporate reports.
South Sudan acting Foreign Minister Peter Bashir Gbandi is set to travel to Ethiopia's capital, Addis Ababa, on Friday to coordinate the mission, the AP news agency reports.
It says that South Sudan does not want Ethiopian troops to go deeper into the country.
The area where the children are being held "is full of jungle", AP quotes Mr Bashir as saying.
Ethiopia shares a long border with South Sudan and cross-border raids are not uncommon.
Hours after news of Friday's attack emerged, Prime Minister Hailemariam Desalegn said the government had requested permission from South Sudan to enter its territory to pursue the attackers.
The South Sudanese ambassador to Ethiopia also said his country was "cooperating and helping" the Ethiopian army to track down the raiders and rescue the abducted children.
Ethiopia's request to enter South Sudanese territory seems to have been approved quickly.
Before South Sudan's civil war began in 2013, joint security operations between the two countries were common as communities on both sides of the border were often involved in cattle raids.
Residents of Gambella town held a demonstration on Thursday demanding "justice for what happened" and calling for better security.
A mother whose husband was killed and three of her children abducted by the attackers earlier told the BBC that she had no hope of seeing her children again.
"I don't know if they were killed during the crossfire," Chol Malual said. "The fighting was intense and if they survived, they will be probably be killed by the Murles."
The targets of the raid were members of the Nuer ethnic group who live in both South Sudan and Ethiopia, the AFP news agency reports.

The 31-year-old is currently playing in the Indian Premier League for Royal Challengers Bangalore, alongside Sussex and England pace bowler Chris Jordan.
Wiese has made 20 T20 appearances for South Africa, including two against England earlier this year.
He will be available to play for Sussex against Somerset on Wednesday, 1 June and Surrey two days later.

The building suffers from flooding, has outdated electrical and drainage infrastructure, is full of asbestos and has fire safety issues.
One of the options being considered will involve both the Commons and the Lords sitting in another location while the Palace is refurbished.
The work, unlikely to begin before 2018 at the earliest, may cost about £1.5bn.
A recent report concluded the cost of maintenance is such that if the Palace was a commercial building of no historical significance it would be considered cost-effective to demolish it and rebuild using modern construction techniques.
A contract to examine the feasibility, costs and benefits of three different renovation options was put out to tender on 15 May.
The successful bidder will advise the parliamentary authorities whether to carry on repairing the building piecemeal as now, attempt a full renovation while Parliament is still sitting in the Palace, or find another venue for Parliament to sit in while the empty Palace is renovated.
Once the contractor is appointed, it will be expected to report within a year.
A report commissioned by the parliamentary authorities in 2012 highlighted the poor state of repair of Britain's seat of government.
Amongst the issues raised were steam leaks which risked distributing asbestos around the Palace and leaking roofs, water pipes and toilets making flooding a "common occurrence".
The building lacks a fire sprinkler system or the sort of fire doors which keep fire contained in a single area, which are common in other commercial buildings. Parliamentary authorities have received advice that some of the cast-iron roofs are in danger of partial collapse if nothing is done.
The report also said that the basements and air-vents are full of electrical and mechanical infrastructure, which is in an uncertain condition. This is because the Palace has been added to on an ad-hoc basis throughout its existence.
The majority of the historic building was built as a replacement for a previous amalgamation of buildings which burnt down in 1834.
The Commons chamber was destroyed in the blitz in 1941 and after some debate it was decided that it should be rebuilt in exactly the same style after the war.
Ethiopia's army has surrounded the area in neighbouring South Sudan where it believes more than 100 abducted Ethiopian children are being held, local media report.
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Parliament has begun a tender process for the renovation of the historic Palace of Westminster.