Taylor said the ninth-year receiver underwent the procedure on Tuesday morning, three days after Green was carted off the field during the team's first practice of the preseason.
ESPN's Adam Schefter initially reported Green suffered a torn ligament in his left ankle. Taylor declined to cite the specifics about Green's injury, but said it was a little more extensive than the team initially thought.
Taylor also didn't provide an exact timetable for Green's return.
"I don't like to make that projection because everybody heals a little bit differently, but we're ready for him to come back when he feels healthy and ready to go," Taylor said. "Hopefully, it's not more than a couple of games."
The first-year NFL coach said the team was initially hopeful that Green could be available for the Bengals' opener on Sept. 8 at Seattle. However, that possibility became unlikely when doctors took a further look at the injury.
Until Tuesday's procedure, Green wore a protective boot in the days following the injury, which was suffered at the University of Dayton's Welcome Stadium.
Green missed seven games last season with a right toe injury, the longest absence of his eight-year career. Before the injury, he averaged 85.9 yards per game.
Since he was drafted with the fourth overall pick in the 2011 draft, Green has been one of the top receivers in the NFL. According to ESPN Stats & Information, he is ranked sixth or higher in receptions (602), receiving yards (8,907) and touchdowns (63).
"Before I got hurt my play has never dipped any," he said last Friday, the day before camp started. "I just happened to have some freak injuries. I always believe in my ability."
The former Georgia standout is at risk of missing multiple games for the fourth time in the last six years.
Green is also entering the final year of his current contract with the Bengals. Before training camp started, he was optimistic he could hit his personal goals for the year if he stayed healthy.
"My number will be high if I play all 16 games," Green said last week. "I know that for a fact."
Immediately after the injury occurred Saturday and he was carted off the field, Green was visibly frustrated and upset.
On Tuesday, Taylor said the team is fortunate that one of the team's top playmakers could still play the majority of the upcoming season.
"We're thankful that it wasn't a lot worse and he'll get a chance to still show all the things that he was setting out to do this season," Taylor said. "He's a fighter."
The Bengals are also without starting wide receiver John Ross, who is officially day-to-day with a hamstring injury. Taylor told reporters on Monday that the injury didn't appear to be a long-term issue.
Tuesday was initially scheduled to be the Bengals' first padded practice of the preseason. However, the team opted to hold a walkthrough indoors after a storm rolled through and delayed practice by more than 90 minutes.
The dampened conditions only added to the team's frustrations surrounding the preseason's rough start.
"It's all about how we respond going forward," Taylor said. "This is just the beginning. It's just day four of camp.
"I'm sure more things will be thrown our way and we're excited to see what those things are."