The Manchester City boss confirmed the approach after the 4-2 FA Cup triumph over Eriksson's Leicester side which booked a fourth round trip to Meadow Lane.
"He tried to get me to manage Notts County," Mancini said. "We spoke about it one or two times but then we finished. When we spoke, it was a good situation. Afterwards it wasn't, so I am happy to be here now."
Little wonder. Mancini was able to name the most expensive starting line-up in FA Cup history at Eastlands, although it took an impressive late strike from Aleksandar Kolarov to see off their npower Championship opponents.
Paul Gallagher had already equalised an early Carlos Tevez effort when Leicester were forced to battle back from two goals down as Patrick Vieira and Adam Johnson scored within 77 seconds of each other.
The visitors took heart from a Tevez penalty miss and when Lloyd Dyer netted eight minutes from time, the Foxes harboured realistic hopes of taking the tie to extra-time.
It was not to be, leaving Eriksson to offer a very positive assessment of his old club.
"I am sure they are one of the teams who can win the league," the former England boss said. "I hope they can win the FA Cup. It would be good for us to have gone out against the winners.
"What is happening at City does remind me of my time at Lazio. At that time we bought Christian Vieri for the biggest transfer fee paid. We sold him 10 months later for even more money.
"Lazio were spending a lot of money. It was a beautiful time. We won the league and overall won seven titles in three years. It was very good because in Italy, if you win the league and you are not Milan, Inter or Juventus, it is a very big thing."