The 43-year-old former player was shot by a man who jumped off a motorbike outside the Dial Bar and Lounge in Santo Domingo at about 8:50 in the evening.
Police have said the operator of the motorcycle carrying the assailant was captured and beaten by a crowd at the bar.
Authorities have both men in custody but have only identified the driver. He is Eddy Feliz Garcia and he was previously arrest for drug possession in 2017.
Ortiz was rushed to the hospital where he underwent emergency surgery.
Tiffany Ortiz stated Big Papi is "resting comfortably" after having a second surgery upon arriving in Boston on Monday.
Ortiz's media assistant, Leo Lopez, revealed a team of surgeons had to "remove part of Ortiz's intestines and colon, as well as his gallbladder", according to Yahoo Sports.
Kennedy said he had been texting Ortiz's wife, Tiffany, who was in the States with their kids, and that "she expressed her gratitude" for all the support.
Doctors said when Ortiz first opened his eyes after surgery, he asked about his family. He was stabilized and then flown to Boston and taken by ambulance, with police escort, to Massachusetts General Hospital.
"There's just one coward out there that just wanted to do that to someone who didn't deserve it".
Leo Lopez, a representative of Ortiz, denied the affair narrative but did say he believed the gunmen had been hired to kill Ortiz.
Another one of Ortiz's former Red Sox teammates, Hall of Fame pitcher Pedro Martinez, who also hails from the Dominican Republic, broke down in tears while talking about Ortiz on Monday night during a broadcast on the MLB Network. "I'd be hard-pressed to think of anyone more beloved than David".
Ortiz galvanized the city in the aftermath of the 2013 Boston Marathon bombing that left three people dead, bellowing through a megaphone at Fenway Park: "This is our (expletive) city!" He was named an All-Star 10 times and helped the Red Sox win World Series championships in 2004, 2007 and 2013.