He is the fourth Reds starting pitcher to be placed on the disabled list this season, joining right-hander Anthony DeSclafani (60-day DL with strained UCL), right-hander Rookie Davis (10-day DL with bruised right forearm) and right-hander Homer Bailey (60-day DL after elbow surgery).
Finnegan left Saturday's game against the Milwaukee Brewers after one inning. The Reds said he had a strained left lat muscle on Saturday but adjusted the nature of his injury Sunday.
"With this particular injury with Brandon, we can realistically expect him to be back here in a few weeks," Reds manager Bryan Price told reporters, according to MLB.com. "But it's not going to be one or two starts and then he'll be back. It's going to be longer than that."
Finnegan walked three of the first four batters he faced, giving up two runs in the inning. The left-hander lasted only two innings in his previous start, finishing with five walks against Pittsburgh.
Romano started and left trailing 3-0 after three innings in his major league debut Sunday, giving up three runs, three hits and four walks, and also hitting a batter. He took the loss as the Brewers beat the Reds 4-2 Sunday for their fifth win in six games.
The Associated Press contributed to this report.