Ortiz hurt his wrist while swinging at a pitch Saturday night in a game against the Baltimore Orioles. He missed Sunday's game, then left the team for Boston, where he underwent an MRI on Monday.
The injury was determined to be a torn ECU tendon shead.
"Surgery is very unlikely," Francona said. "We'll certainly have more information [Tuesday].
The move to place Ortiz on the disabled list will occur Tuesday, Francona said, and Boston will recall right-hander Justin Masterson from Triple-A Pawtucket to fill Ortiz's spot on the roster. Masterson will start Tuesday night against Tampa Bay.
The Red Sox were hopeful Ortiz would not be gone for long.
"I don't want to speculate," Francona said. "All of us want to get a chance to talk to David and the doctors. We'll map this out to the best of their ability. I don't know more. I can't tell you more. ... We need to talk about some things."
Ortiz is batting .252 and leads the Red Sox with 13 homers and 43 RBIs.
"We've got to keep playing, the same way we're playing until he gets back," left fielder Manny Ramirezsaid. "He's a big part of the team. What can you do? You don't plan to get hurt. Let's be patient and wait for him."
Francona has options to substitute for Ortiz.
"We have good hitters," Francona said. "There are some things we can do. We can move Manny to the DH, and have an outfield that runs all over the place. We'll get it figured out."
Ramirez replaced Ortiz as Boston's DH on Sunday and Monday. He said he relishes the role.
"I love DH," Ramirez told the Boston Herald. "I'm pretty sure they're not going to miss my Gold Glove out there [in left]. We're fine. We're going to be fine."
Information from The Associated Press was used in this report.