The 35-year-old slugger hit 25 homers in 428 plate appearances for the Indians and Mariners in 2010.
Branyan would give the D-backs another option at first base. Currently they have newcomer Juan Miranda and Brandon Allen competing for playing time. Other candidates include Xavier Nady and Geoff Blum.
General manager Kevin Towers acquired Miranda from the Yankees this winter after getting an up-close look at him while serving as a special assistant to New York GM Brian Cashman last year.
Branyan has played for eight teams during a 13-year career and played for the Padres in 2006 and 2007, when Towers was San Diego's GM.