The Canaries hope to announce Daniel Farke's successor within the next 72 hours after relieving the German of his duties at Carrow Road after their victory against Brentford.
Smith was only sacked by Aston Villa last weekend after five defeats on the trot in the Premier League.
Norwich had been keen on appointing Frank Lampard but the ex-Chelsea boss turned down their advances after consultation with friends and former colleagues.
According to reports, Norwich were imressed with the former Walsall and Brentford boss after an interview where he boasted 'an in-depth knowledge of the squad, playing style and culture of the club'.
The Canaries will be hoping to sort their current managerial situation out as quick as possible, as they take on Southampton at Carrow Road after the international break.
Norwich have won just five points so far this season - the same as Newcastle - but sit in last place due to their goal difference.
They sacked ex-boss Farke despite claiming their first win of the season last weekend, which gave them some hope in their bid for Premier League survival.
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