The Boston Globe's Pete Abraham wrote Friday that "an announcement is expected in the coming days," and cited sources as saying Ortiz is seeking $15 million for the 2015 season.
"Everything is going the right way," Ortiz told the paper this week. "I'm not worried."
The veteran designated hitter batted .308 with 30 home runs and 103 RBIs last season.
According to the Globe, Ortiz's agent, Fernando Cuza, has had at least three in-person meetings with the Red Sox during Spring Training.
Joey Nowak is a reporter for MLB.com. Follow him on Twitter at @joeynowak. This story was not subject to the approval of Major League Baseball or its clubs.