TODAY, 2:48pm: Even the reliever-rich Royals have expressed interest, Ken Rosenthal of FOX Sports tweets.

11:05am: The Dodgers are “showing significant interest” in O’Day, Jerry Crasnick of ESPN.com reports (Twitter links). Jon Heyman of CBSSports.com previously connected O’Day to the Dodgers. With a number of teams pursuing the veteran, some executives tell Crasnick that a deal well in excess of last year’s Luke Gregerson contract (three years, $18.5MM) could be in order.

MLBTR’s Tim Dierkes has predicted that O’Day will take down $22.5MM over three years. His age (33) is perhaps the only major limiting factor in his market, and a serious run-up in demand could conceivably get him a fourth year guaranteed.

YESTERDAY: The market for free-agent right-hander Darren O’Day is moving quickly, reports ESPN’s Buster Olney (links to Twitter). The Beverly Hills Sports Council client already has offers in hand, according to Olney, and he’s planning to meet with the managers of some of those teams in the coming week. Olney lists the Tigers, Red Sox and Giants as three clubs to have shown interest in O’Day, though it’s not known if any of those three are among the teams to have extended a formal offer.

Typically, it behooves free agents — especially relievers — to sign contracts early in the offseason. A look back to November relief signings over the past few years shows the likes of Zach Duke (three years, $15MM), Joe Smith (three years, $15.75MM), Javier Lopez (three years, $13MM), Jonathan Broxton (three years, $21MM), Jeremy Affeldt (three years, $18MM) and Joaquin Benoit (three years, $16.5MM back in 2010) all landed sizable contracts despite not being considered the top names on the free agent market. Of course, O’Day also has the advantage of being the consensus top name on this year’s relief market, so perhaps it’s not as crucial for him to sign early. Still, striking early while each club has its entire offseason budget at its disposal isn’t a bad route for O’Day’s camp.

Over the past four seasons, the 32-year-old O’Day has worked to a 1.92 ERA with 9.7 K/9 and 2.1 BB/9. He saved his best season for his contract year, establishing new career-bests in ERA (1.52) and K/9 (11.3); opponents batted just .198/.257/.283 against O’Day. He yielded just a .210/.293/.333 batting line to left-handed hitters and dominated right-handed bats, who mustered only a .192/.236/.256 line against him.

Interest in O’Day was said to be “steep” from day one at the GM Meetings, with the Cubs, Royals, Nationals and Dodgers all being connected to the former Oriole yesterday as well.