The Rangers and Sosa's agent, Adam Katz, have exchanged a couple proposals since the 38-year-old, who has 588 career homers, worked out twice for Texas. The most recent workout was Jan. 15.
An agreement appears likely, but Katz has been traveling and the wife of Rangers general manager Jon Daniels is due to give birth to the couple's first child.
Texas did agree to a minor-league contract with right-hander Jamey Wright, who will go to spring training with a chance to compete for the fifth spot in the rotation behind Kevin Millwood, Vicente Padilla, Brandon McCarthy and Robinson Tejeda.
Wright has a 67-98 career record with a 5.14 ERA in 260 games (237 starts) over parts of the last 11 seasons with Colorado, Milwaukee, St. Louis, Kansas City and San Francisco, where he was 6-10 with a 5.19 ERA in 34 games (21 starts) last season.
If Wright is placed on the 40-man roster, he would get a $900,000, one-year contract and the chance to earn $350,000 in performance bonuses based on innings. The bonuses start at 120 innings and top out at 180 innings.
Like Wright, Sosa will have to earn his spot on the Rangers' roster.
Sosa hasn't played in the majors since 2005 with Baltimore, when the seven-time All-Star hit .221 with 14 homers and 45 RBIs in 102 games. His major-league debut came with the Rangers in 1989, when he played only 25 games before being traded to the Chicago White Sox.
If completed and he makes the team, Sosa's contract would likely have a base salary of about $500,000. He would have the chance to earn additional performance bonuses.
In his career with the Rangers (1989), White Sox (1989-91), Chicago Cubs (1992-2004) and Orioles, Sosa has batted .274 with 1,575 RBIs and has hit 60 home runs or more in a year three times.
Sosa was the NL MVP in 1998, when he batted .308 with a career-high 66 homers and 158 RBIs for the Cubs. That was the season he was in the home run chase with Mark McGwire, who became the first major-leaguer to hit 70 homers.