Simmons, 26, hit .265 with four homers and 44 RBIs in 535 at-bats in 2015. He was beaten out by Brandon Crawford for the National League Gold Glove at shortstop.
Simmons is signed through 2020 at salaries of $6 million, $8 million, $11 million, $13 million and $15 million.
Aybar, a 2014 All-Star and Gold Glove winner in 2011, is a 10-year veteran who has spent his entire career with the Angels. He hit .270 with three home runs and 44 RBIs in 2015.
Sources said the two minor league pitchers heading to Atlanta are Sean Newcomb and Christopher Ellis. Newcomb, a 6-foot-5 left-hander, was rated the No. 25 prospect in baseball in ESPN Insider Keith Law’s midseason Top 50 in July.
The Braves had earlier been engaged with the Mets on a potential deal for Simmons, including as recently as Thursday morning at the GM meetings in Florida.
Information from ESPN’s Buster Olney, Jerry Crasnick and Adam Rubin was used in this report.