None of the moves had been formally announced by the Reds.
Cueto, 28, finished 2014 with a 20-9 record and a 2.25 ERA in 34 starts. A first-time National League All-Star, he led the league in starts and innings (243 2/3), and he was tied for first in strikeouts with 242.
Ludwick, 36, was originally signed as a free agent in 2012 and hit 26 home runs with 80 RBIs in his first season in Cincinnati. His production declined to 11 home runs and 57 RBIs over the past two seasons following a shoulder injury suffered on Opening Day 2013 while sliding head-first into third base. He missed most of that season on the disabled list after having surgery.
This season, Ludwick batted .244 with nine homers and 45 RBIs in 112 games.
Shoulder issues also marred Hannahan's production while in Cincinnati. In only 109 games over two seasons, he batted .209 with one homer and 16 RBIs. Hannahan played with a torn labrum in his right shoulder during 2013 and had surgery last October. A natural third baseman, he was activated in late July and was limited to playing some first base pinch-hitting.
In another move, the club also sent infielder Neftali Soto outright to Triple-A Louisville. Soto batted .100 (3-for-30) in 21 games over two big league stints this season.
Mark Sheldon is a reporter for MLB.com. Read his blog, Mark My Word, and follow him on Twitter @m_sheldon. This story was not subject to the approval of Major League Baseball or its clubs.