The most likely candidates are left-handers Erik Bedard, Cesar Ramos and Mike Montgomery along with right-hander Nathan Karns.
"We're still talking about it," Rays executive vice president of baseball operations Andrew Friedman said. "The fact is right now we're essentially down three of our starters that we were looking at coming into the year," referring to Alex Colome (suspended), Jeremy Hellickson (right elbow) and Moore.
"We'll work through it. It's what we always talk about why depth is so important. We have some guys who we feel will give us a chance to win every fifth day, so we'll continue the conversation to see which one makes the most sense on Sunday."
Beliveau learned about Moore's injury on Monday night.
"People were talking once we were in the clubhouse from our game," said Beliveau, who did not pitch Monday. "I actually went to my house and was watching the national championship game with Karns. And we were kind of looking stuff up and playing GM like every player does. I didn't really think much of it. I got a call from [Durham manager] Charlie [Montoyo] this morning at 8:30 saying, 'You're on an 11 o'clock flight.'"
The Rays acquired Beliveau from the Rangers in exchange for cash considerations on April 16 of last season. He had four tours with the Rays but did not appear in a game until his last one of the season. He was up for a total of four days in his first three stints.
The left-hander does not know how long he will be around, but he feels good about the way he has been pitching.
"Toward the end of Spring Training I felt really well," Beliveau said. "Getting all my pitches together. Throwing the other day I threw really well. All my pitches were strikes."
Bill Chastain is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.