The Spanish publication says in its Friday's front cover (16 February) that Iniesta is yet to make a decision but the Chinese outfit are currently "optimistic" having made him a stunning formal offer which could worth up to €35m (£31m, $43.8m) per season – equivalent to £596,000-a-week.
Iniesta's future at the Nou Camp was under question during the last summer transfer window after he saw his playing time restricted during Luis Enrique's last season in charge.
The 33-year-old midfielder was due to be out of contract at the end of the current season and there were suggestions that he could follow the same path of Xavi Hernandez - who has moved to Qatar to play for Al Sadd - and continue his career in a minor league.
However, Iniesta managed to turn around his fortunes under Ernesto Valverde and in October decided to put pen to paper on a lifetime contract at Barcelona.
There were suggestions that the Spain World Cup winner could see his playing time limited following the big money arrival of Philippe Coutinho from Liverpool during the January transfer window.
Yet, Iniesta has since remained unmoved in Valverde's plans whenever he has been fit, starting a number of key games including in both legs of the Copa del Rey semi-final against Valencia.
But Sport says that the veteran midfielder is once again mulling over his future at the Nou Camp after having received an offer to become one of the best paid players in the world.
The report says that Tianjin Quanjian first contacted Iniesta in December and representatives from the Chinese club have twice been to Spain to try to and convince the Barcelona legend to make the move.
Iniesta ruled out leaving the La Liga giants in January during the first meeting but Sport says that he is now considering making the move in the summer.
Sport says that Iniesta will wait until the end of March to make a decision but Barcelona officials are already aware of the offer and believe it will be difficult to keep him.
Furthermore, the Spanish publication claims that, if the player decides to move, the club will respect his decision as they already did in with Xavi and more recently with Javier Mascherano, who moved to Hebei China Fortune last month.
Iniesta joined Barcelona at the age of 12 in 1996 and has since become a legendary figure, helping them win 30 major honours, including eight La Liga titles and four Champions League crowns.