In a statement, the attorney general's office named the alleged mastermind behind the shooting as Alberto Miguel Rodriguez, who is a fugitive from justice.
Ortiz was shot in the back on June 9 while he was in a nightclub in his hometown Santo Domingo.
Doctors removed his gallbladder and part of his colon and intestines, and he also sustained liver damage.
Ortiz, 43, was first treated in his native Dominican Republic and then flown to Boston last Monday for a second operation.
So far 10 people have been taken into custody over the attack, including one who surrendered on Friday, and another four are on the run, including Rodriguez.
The alleged trigger man is among those in custody. Police say he is a 25-year-old who claims to have been offered the equivalent of around $8,000 to shoot Ortiz.
A judge has ordered that nine of the suspects will remain in custody for up to a year while the investigation proceeds, as is allowed under Dominican law.
The motive for the attack on Ortiz remains unknown.
The former designated hitter and first baseman, who retired in 2016, was in the Dominican Republic for business and personal reasons.
Known affectionately as "Big Papi," he played 14 seasons for the Red Sox and made 10 All-Star appearances in his 20-year career.
Ortiz hit 541 home runs with 1,768 RBIs in 2,408 games in the major league.
Ortiz began his career by playing six seasons for the Minnesota Twins (1997-2002), but his career took off after he joined the Red Sox.
He helped Boston capture their first World Series title in 86 years in 2004, when he was the MVP of the American League championship series.