Rea, who was acquired from the Padres on Friday, exited his first start with Miami in the fourth inning of Saturday night's 11-0 win. The right-hander allowed one hit over 3 1/3 innings with four strikeouts.
"It still feels the same today as it did last night," Rea said on Sunday morning. "We still haven't done anything yet, just kind of waiting right now."
The DL stint was expected, and Rea was informed of the club's intentions on Saturday.
"They kind of told me after the game what they were going to do," Rea said. "Now, we're just going to wait to get it evaluated and get some more tests."
It isn't immediately clear who will take Rea's spot in the rotation, but Ogando will likely provide some bullpen depth. In three appearances with the Marlins this year, he owns a 3.00 ERA in three innings.
Ogando spent some time with the Marlins before he went on the DL with a fractured rib on May 10.
"It wasn't easy coming back from breaking a rib," Ogando said. "I took like two months here, playing a little catch, but I started feeling good after like three weeks of playing games."
When the Marlins acquired Rea and right-hander Andrew Cashner on Friday, they were hopeful that they solidified their rotation for the stretch run. But Rea was injured just 44 pitches into his Marlins' debut.
Now, the club is re-engaged in the market for a starter prior to Monday's 4 p.m. ET non-waiver Trade Deadline.
After wrapping up their 10-game homestand on Sunday, the Marlins will face the Cubs on Monday for the first of three games at Wrigley Field.
The Marlins will go with Adam Conley on Monday, Jose Fernandez (Tuesday) and Tom Koehler (Wednesday). The club is off on Thursday before traveling to Colorado for three games with the Rockies.
If the Marlins don't swing a trade, they have a few internal candidates, like Triple-A New Orleans lefty Justin Nicolino and right-hander Jose Urena. At Double-A Jacksonville, Jake Esch is a possibility.
Although Urena was optioned to New Orleans on Saturday after he started on Friday, he could be called back sooner than the required 10-day period if there is a DL situation.
David Phelps is another possibility. The right-hander has mostly been used in setup relief this season, and he has already thrown 54 1/3 innings in 50 appearances. But Phelps has made 59 starts in his career, including 19 last year with the Marlins, and he is 16-19 with a 4.36 ERA in that role.