Ortiz arrived in the clubhouse at spring training in Fort Myers, Fla., on Monday morning and sought out Price.
"I got your back. Let's roll," Ortiz told Price after they embraced.
Thus, Ortiz and Price have officially ended any feud from their past.
During the 2013 American League Division Series, Price was upset that Ortiz took so much time circling the bases after a pair of home runs.
Then in 2014, Ortiz declared "war" between the two players in a postgame interview after being hit by a pitch from Price.
Ortiz said then: "I mean, it's a war. Next time he hits me, he better bring the gloves. I have no respect for him no more."
Price was offended by Ortiz's depiction of the incident as "war."
"We have troops, men and women that are fighting for our freedom, making sure we stay safe and could be prepared for a real war," Price said then. "That's not acceptable. This is not a war, by no means. This is a game we play."
The Red Sox signed the 30-year-old left-handed Price to a seven-year, $217 million contact during the offseason.
''I'm ready to be his teammate and his friend,'' Price told reporters of Ortiz on Sunday. ''That's what I'm looking forward to. I want to be accepted by all my teammates and that's something you need to have if we want to get to where we need to be.''
Price is slated to receive $30 million annually from 2016 through 2018, $31 million in 2019 and $32 million from 2020 through 2022. The five-time All-Star is 104-56 with a 3.09 ERA in 218 career games (213 starts) with the Tampa Bay Rays, Detroit Tigers and Toronto Blue Jays.
The 40-year-old Ortiz announced in November that he will retire after the 2016 season.
At the end of his first spring training workout of his final major league season, Ortiz hopped into a golf cart with fellow Red Sox legends Jim Rice and Luis Tiant, according to MLB.com.
"This is going to be me next year -- chilling," Big Papi said of the ride.
This year will be Ortiz's 20th in the major leagues. He has spent the past 13 seasons with the Red Sox after six years with the Minnesota Twins at the start of his career.
Ortiz, a nine-time All-Star and a six-time Silver Slugger Award winner, finished last season with a .273 average, 108 RBIs and 37 home runs.
The popular, Dominican-born Ortiz joined the 500-home run club last season and finished with 503 career homers, 1,641 RBIs and a .284 lifetime batting average. He has hit 445 of his homers since joining the Red Sox in 2003. His 447 home runs as a designated hitter are the most in history.
Ortiz led the Red Sox to three World Series championships in 2004, 2007 and 2013. He owns the best World Series batting average (.455), on-base percentage (.576) and slugging percentage (.795) for players in the Fall Classic. He was named MVP of the 2013 World Series.
"He's in great shape," manager John Farrell told MLB.com. "I've had a chance to sit with him a couple of times in the offseason. I know he's looking forward to this year for a number of reasons. But above all, he wants to win, as we all do, and I think it speaks volumes the way he showed up, the condition that he's in. We look forward to him being a main cog and a main presence in our lineup."