Ljungerg, who has spent nine years with the Gunners, has been told he can leave the Emirates Stadium by Arsene Wenger.
Fiorentina were thought to be close to finalising a £5.5million deal for the 30-year-old but The Sun claims Eriksson has now held talks with Ljunberg and his agent.
The paper reports the Eastlands outfit are now favourites to land the midfielder for £6million after a successful meeting on Tuesday night.
Ljungberg is on a hefty wage at Arsenal but Eriksson has been told he has a £50million warchest to play with and he is keen to spend it on a player he named the best Swedish footballer still playing.
City are also close to signing Reggina striker Rolando Bianchi for £8.8million.