The Canaries hope to announce Daniel Farke's successor within the next 48 hours after relieving the German of his duties last week.
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.
Now Smith is poised for the hotseat at Carrow Road just days after leaving Villa.
Smith, 50, was in charge of Villa for three years and led them from the Championship into the Premier League.
He kept them up in their first season back in the top-flight and led them to an 11th place finish last season, while they also reached the Carabao Cup final in 2020.
Smith is understood to be eager to make an instant return to management and will now be handed that chance by Norwich.
The Canaries want the new manager in place ahead of their clash with 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 against Brentford.