Iniesta left Barcelona after 22 years with the club at the end of the 2017-18 campaign, joining Japanese side Vissel Kobe on a three-year deal.
Barca boss Ernesto Valverde is understood to have made signing a new midfielder one of his priorities for the current transfer window, and the Daily Star reports that Gundogan is near the top of his list.
The 27-year-old joined Man City in 2016 but missed the vast majority of his first season with the club due to a serious knee injury.
However, he made 48 appearances across all competitions during City's title-winning campaign last term, and his form during that time has reportedly convinced Barcelona to make a £50m approach for the Germany international.
City have been heavily linked with a move for Napoli's Jorginho this summer and could be tempted by the big-money offer, which is more than double what they paid for Gundogan two years ago.