Doctor Abel Gonzalez believes Ortiz's condition is becoming stable enough to make it possible for him to travel to Boston, and his fellow doctors have all approved the patient's trip to proceed as soon as the plane is ready.
Ortiz, a fearsome hitter, is one of the most beloved figures in sports history in the Dominican Republic.
"On behalf of me and my family, I want to thank (owners) John and Linda Henry, Tom Werner, (president) Sam Kennedy, and the Boston Red Sox for all that they are doing for David and our family, as well as Dr. Larry Ronan and the fantastic staff at Massachusetts General Hospital".
In 20 years in the Majors, Ortiz hit 541 homers, including 14 with the Red Sox. The gunman has not been arrested, and police say they do not yet have a motive.
Authorities said Ortiz was ambushed by a man who got off a motorcycle and shot Ortiz in the back at almost point-blank range around 8:50 p.m. local time at the Dial Bar and Lounge.
Also wounded in the attack were the Boston baseball legend's friend, TV host Jhoel Lopez, and a third person.
Ortiz, 43, "underwent a successful second surgery", his wife, Tiffany, said in a statement released through the Red Sox on Tuesday.
He spent six hours in surgery overnight, with doctors removing his gallbladder and parts of his intestines and colon to stem the bleeding, officials said. "A lot of times I see guys who play on my team coming to the United States to live". "Many people are losing their lives".
Former Red Sox teammate Pedro Martinez tweeted out a photo of himself hugging Big Papi, writing, "I'm at peace knowing you're out of danger". "Thank God no amount of evil or hate can overcome a heart that big".
"We ask for privacy as David works toward recovery".
Obama referenced Ortiz's famous speech at Fenway Park in April 2013 ("This is our f--ing city!") just days after the Boston Marathon bombings that helped unite the city in a time of distress. Fans responded to the shooting with an outpouring of prayers and wishes for a speedy recovery.