Wright, who turns 30 next month, is 10-18 with a 4.17 ERA in seven major league seasons, the first 5 1/2 with the Astros. He was 0-3 with a 3.17 ERA in 58 games this season with the Cubs, who non-tendered him last month.
Wright can become a free agent following the 2015 season.
Wright, a seventh-round pick of the Dodgers in the 2003 First-Year Player Draft, has held left-handers to a .238/.315/.331 slash line. Right-handers are batting .264/.354/.479 against him.
The Orioles lost Andrew Miller to free agency and have made Brian Matusz available in trade talks, necessitating at least one left-handed addition to their bullpen.
Wright faced the Orioles on Aug. 23 at Wrigley Field and tossed 1 2/3 scoreless innings.