Chelsea owner Roman Abramovich is widely expected to initiate a summer turnover of his rapidly aging squad.
Drogba's expiring contract makes him among the most likely Blues superstars to turn up elsewhere next season.
Barcelona might be more impressed with the aging forward than most because of the threat he's posed in recent years when the Catalans met Chelsea.
Drogba has scored three times in the teams' last six meetings, including the only goal in the first leg of last month's Champions League semifinal.
The 34-year-old has scored 11 goals in 31 appearances for Chelsea this season. Five of those came during seven appearances in the all-important Champions League.
Barcelona are reportedly prepared to offer Drogba a one-year contract with an option for a second.
For the player's part, the Spanish news outlet is reporting that he recently told a journalist he would love to play alongside Lionel Messi, Xavi Hernandez and Andres Iniesta.
From a competitive standpoint, the Barcelona rumor is by far the most interesting one linked to Drogba.
Other rumors have the player moving to Qatar, Russia or China in deals that would clearly be most attractive for their financial terms.