Burnett, 37, became a free agent when the Pirates failed to extend him a $14.1 million qualifying offer in December. He made $16.5 million last year ($8.5 million of that was paid by the New York Yankees).
Since he's older and only recently considered retirement, Burnett likely could be had on a short-term deal, the kind of low-risk contract that could fit into the Phillies' plans.
The Phils reportedly pursued Masahiro Tanaka, at least on the periphery, and budgeted him into their plans. Perhaps ownership would make a similar exception for Burnett.
Although Amaro sounded finished with his starting rotation, manager Ryne Sandberg didn't rule out any additions to his staff before the team reports to camp in 2 weeks.
"We'll see what happens," Sandberg said of his rotation, which is full of question marks after Cole Hamels and Cliff Lee. "Whether we're done [making additions] or not, that remains to be seen. Something could still be done."
Just as recently as last week, ESPN.com reported that the Phillies and Orioles had shown interest in Burnett this winter.
When the winter began, Burnett seemed like an ideal fit for the Phillies as a replacement for fellow free agent Roy Halladay.
Burnett was 26-21 with a 3.41 ERA in 61 starts with the Pirates in the last two seasons. In 2013, when he had a 3.30 ERA in 30 starts, Burnett led all National League starters in strikeout rate and groundball rate, which would seemingly make him a very nice fit at home run-friendly Citizens Bank Park.
As a righthander, Burnett could also slide into one of the top spots in the rotation if the Phillies wanted to break up their two lefthanders.
Burnett also has a connection to one of the Phillies' chief free-agent negotiators. Burnett is a neighbor and friend of Phils assistant general manager Scott Proefrock; both live in suburban Baltimore. Burnett's agent is Darek Braunecker, who also represents Cliff Lee.
Three winters ago, Proefrock and Braunecker feverishly (and quietly) worked on the free-agent deal that brought Lee back to Philadelphia.
At the winter meetings last month, Amaro said he "left no stone unturned" when he was asked whether he had pursued Burnett. When those same meetings ended, however, Amaro filled out his rotation when he signed Roberto Hernandez to a 1-year, $4.5 million contract.
But after taking care of their arbitration players in the last 2 weeks, the Phillies should have the financial flexibility to add a pricey player. Their current Opening Day payroll figures to be in the $165 million to $170 million range, well below the $189 million luxury-tax threshold.
Additionally, the Phillies, along with every team in baseball, received $24 million this offseason as a part of MLB's new national television contract.
If a last-minute move for a pitcher fits into the payroll, does it fit into the pitching staff?
The current rotation features three locks - Hamels, Lee and Kyle Kendrick - and since Amaro signed Hernandez to be a starter, we'll up that to four for argument's sake.
Cuban import Miguel Alfredo Gonzalez (in the first year of a 3-year, $12 million deal) would figure to have an edge on Jonathan Pettibone for the fifth spot, but neither is guaranteed the job, either. Some have even suggested that Gonzalez, who hasn't pitched competitively in 2 years, could even wind up as a late-inning reliever.
If the Phillies are interested in Burnett, they won't be alone. Since the Pirates did not make Burnett a qualifying offer, a team that signs the pitcher would not have to forfeit a first-round pick as compensation.
Burnett's timing also couldn't be better, for his own sake: after a lull in the market, Tanaka signed with the Yankees last week and fellow righthander Matt Garza followed shortly thereafter, signing with the Milwaukee Brewers. Teams that missed out on those pitchers are likely to inquire about Burnett.
With 2 weeks remaining before most teams open their spring-training camps, Burnett isn't the only pitcher still looking for work. Among other free agents: Ubaldo Jimenez, Ervin Santana and Bronson Arroyo.
On Twitter: @ryanlawrence21