But with the realization that the cost to acquire Greinke will be steep, the Braves are evaluating other options and continuing to monitor the possibility that they might still land Ryan Dempster.
There is a chance the Braves will add a reliever and possibly a utility player to strengthen their bench before Tuesday's 4 p.m. ET Trade Deadline. But the primary objective is to add a front-line starting pitcher like Greinke or Dempster.
"We're looking to continue to improve," Braves general manager Frank Wren said. "We'll see how that shakes out in the next couple of days."
Atlanta has interest in Tampa Bay right-hander James Shields. But like the Marlins with Josh Johnson and the Padres with Edinson Volquez, the Rays are providing every indication that they will not move him unless they are blown away with an offer.
It has been an interesting week for the Braves. When Atlanta agreed to a deal to acquire Dempster on Monday, the Brewers still seemed to be holding out hope of signing Greinke to a contract extension.
Since then, Dempster has used his right to block a trade to the Braves and the Brewers have more reason to deal Greinke after suffering three consecutive heartbreaking losses to the Phillies.
Brewers GM Doug Melvin told WTMJ Radio on Thursday morning that he has been fielding calls regarding Greinke, who is scheduled to make his next start on Sunday. The most interested clubs appear to be the Braves, Angels, Rangers and White Sox.
The Braves were willing to trade highly-regarded pitching prospect Randall Delgado and at least one other Minor League player to the Cubs for Dempster. The cost to get Greinke would be more expensive from a prospect standpoint.
The Brewers have indicated a deal would require the Braves to part ways with top prospect Julio Teheran and at least one other highly-regarded farmhand.
If Greinke were to exit after becoming a free agent at the end of this season, he would an extremely expensive two-month rental. Five years after mortgaging their future for one calendar year of Mark Teixeira, the Braves will approach these negotiations cautiously.
The Braves would like to keep Greinke beyond this season and will likely bid for him this winter even if they do not acquire him over the next five days. But the former American League Cy Young Award winner's cost will be influenced by the six-year, $144 million contract extension Cole Hamels received from the Phils on Wednesday.
While Dempster has impressed while compiling a 2.25 ERA this year, Greinke is more attractive because of his ability to serve as a legitimate ace who would have the potential to match up against the top-flight pitchers the Braves would face if they reach the playoffs.
Still, Dempster could be quite valuable to a Braves rotation that has proven inconsistent throughout this season. Its most dependable members right now are 37-year-old Tim Hudson and Ben Sheets, who was retired for nearly two full years before Wren's staff made the wise decision to sign him to a Minor League contract earlier this month.
Wren said on Wednesday that he has "moved on" from Dempster, and that it was "highly unlikely" that the veteran right-hander will end up in Atlanta. But he would not completely rule out the possibility.
Dempster has made it known that his preference is to be traded to the Dodgers. This would reunite him with his close friend and former Cubs teammate Ted Lilly. But he has not completely ruled out the possibility of coming to Atlanta if a deal with the Dodgers does not materialize.
When Dempster was making his way through the Minor League levels with the Marlins, current Atlanta skipper Fredi Gonzalez was one of his managers and Wren was one of the Marlins' assistant general managers.
"I respect Frank Wren a lot," Dempster told reporters on Wednesday. "I've known him for a long time. He was an assistant GM in Florida, and I know Fredi and those guys, but they also have a job to do and that's to better their team, and they can't wait around for me to make my decision. I understand that fully. If that's still an option down the road, it's something I'll have to look at. I'll try to make the best decision for me and my family."
The Dodgers were unable to reach an agreement when they discussed acquiring Dempster earlier this month. It is believed they have more interest in pursuing Matt Garza, another Cubs pitcher.
"It's not over," Chipper Jones said of the chance to get Dempster. "I'm sure when you've been in one spot for a really long time, being traded is a big adjustment and there is more than just the business of baseball at hand."
Mark Bowman is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.