Following the decision, Cora said he told Rodriguez: "The way you're throwing, at some point in your career, you're going to be an Opening Day starter—maybe more than once."
Nathan Eovaldi will instead take the ball when the Red Sox open their 2021 season Thursday against the Baltimore Orioles at Fenway Park.
Rodriguez missed the entire 2020 campaign after testing positive for COVID-19 in early July and then being diagnosed with myocarditis, an inflammation of the heart wall that can be a long-term complication found in people who've had the coronavirus.
The 27-year-old Venezuela native returned to the club for the start of spring training last month.
"At the beginning of COVID, I just wanted a chance to get past through that," Rodriguez told reporters. "Thank God I have a chance to survive that part and be available to get back to baseball and get cleared. ... I did everything that I can to get back to 100 percent, and now I'm here. I'm ready to go."
Cora confirmed Rodriguez's current situation is not linked to COVID-19 or myocarditis, and the Red Sox are hopeful he's able to make a quick recovery.
"Next step is to throw a bullpen," Cora said. "So when he throws a bullpen, then we address the situation again."
Rodriguez finished sixth in AL Cy Young Award voting in 2019 when he posted a 3.81 ERA with 213 strikeouts in 203.1 innings across 34 starts for Boston.
Meanwhile, Eovaldi has flashed occasional brilliance thanks to a fastball that sits comfortably in the upper-90s, but he's lacked consistency during his 10-year MLB career.
The 31-year-old right-hander showed signs of progress last year with a 3.72 ERA, 1.20 WHIP and 52 strikeouts in 48.1 innings. His strikeout-to-walk ratio (7.4) was the best mark of his career, albeit in an abbreviated campaign where he made just nine starts.
Boston is looking to rebound from a disappointing 2020 season where it finished with a 24-36 record, one game behind the Orioles from last place in the AL East.
Any type of extended absence for Rodriguez would be a major blow to those bounce-back efforts.