 Eriksson was linked with the Fulham job in the summer
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The ex-England boss had been linked with the job after Paulo Sousa was sacked on Friday with the Foxes bottom of the Championship after nine games.
Eriksson, 62, is believed to have been offered a two-year deal to take charge.
The Swede watched from the directors’ box at the Walkers Stadium as City beat Scunthorpe 3-1 on Saturday, however he deflected questions from the media.
Both coach Chris Powell and goalkeeping coach Mike Stowell, the duo who took charge of the first team for the victory over the Iron which lifted Leicester off the foot of the table, expressed a wish to work for Eriksson.
City’s Swiss left-back Bruno Berner, meanwhile, was buoyed by Eriksson’s anticipated arrival, although the 32-year-old spared a kind word for the departed Sousa.
“He is a former England manager and I see only positives in this signing,” said Berner.
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“We did not win enough games we were not effective enough, at the end we conceded too many goals and you could see the decision [Sousa’s sacking] coming but we were all surprised by the timing.
“I personally learned a lot from him because he was a very knowledgeable manager.”
Eriksson, a former Manchester City manager, has been out of work since managing the Ivory Coast at this summer’s World Cup in South Africa, where they failed to reach the knock-out stages.
Prior to that, he had a seven-month spell as Notts County’s director of football having managed Mexico’s national team from June 2008 to April 2009.
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Leicester, who were recently taken over by a Thai consortium, reached the play-off semi-finals last season under Nigel Pearson, who subsequently left to take charge at Hull City.
On the back of taking Swansea to the brink of the Championship play-offs last season, former Portugal international Sousa was then recruited by chairman Milan Mandaric but oversaw a woeful start to the season, with only five points gathered from Leicester’s first nine league matches.
A Carling Cup run including away wins over Leeds and Portsmouth provided some relief.
But with many fans questioning Sousa’s tactics and team selection, his fate was sealed by a 6-1 league loss to Pompey, followed by a 4-3 defeat at Norwich.
If Eriksson’s appointment is confirmed, the former Lazio boss will become Leicester’s seventh manager since former Portsmouth owner Mandaric took over in 2007.
