NASHVILLE — Brett Lawrie’s time in Oakland appears to be over.

Acquired from the Blue Jays in the Josh Donaldson deal last year, Lawrie was close to being moved to the White Sox for two minor leaguers late Tuesday night. According to two big-league sources at the winter meetings, the teams were still discussing names for the return portion, but the Lawrie part was essentially settled.

The Chronicle also has learned that the A’s are moving in on free-agent reliever John Axford as the fourth addition of the winter to the bullpen, which was the worst in the American League last season.

Earlier in the day, A’s vice president of baseball operations Billy Beane had joked his team might have made an even bigger splash at one point.

“We were right in that Greinke thing until we found out it was actually just one year for $35 million,” Beane cracked of the $206.5 million, six-year deal starter Zack Greinke signed with Arizona. “They said, ‘Thirty, 35 (million),’ and I said, ‘We’re in!’ And they said for one year. Per year. Per annum.”

The payroll will go up next season, Beane said, but there are still limitations on what the A’s can do when it comes to jumping into the increasingly high-priced market for starters.

The team has had talks with former A’s left-hander Scott Kazmir, who is looking for a deal in the $50 million range, but another onetime Oakland pitcher, Bartolo Colon, might be more in the team’s price range should the dollars get too high for someone such as Kazmir.

Oakland has hit the remainders bin a lot in recent years, signing players who’ve missed significant time with injuries and even stepped away from the game or gone to independent ball (Kazmir, Rich Hill, Ryan Madson, even Frank Thomas and John Jaha, as Beane noted). But 2008 Cy Young winner Cliff Lee, who is gaining attention from numerous teams as he tries to make a comeback from elbow problems, is likely to cost more than the A’s wish to spend.

The team’s bullpen revamp continued, with the team sending right-hander Evan Scribner to the Mariners late Tuesday and with the Madson acquisition pending later in the week. And there’s more to come: Beane said that the A’s could add another reliever even after the addition of Madson, whose three-year, $22 million deal will be announced after he has a physical Thursday.

Scribner is out of options and is coming off a rough year in which he allowed 14 homers, the most among major-league relievers. Oakland received minor-league reliever Trey Cochran-Gill, a 22-year-old right-hander who went 6-5 with six saves and a 4.18 ERA in 45 games at three levels of the Mariners’ organization. He didn’t allow a homer in 751/3 innings.

Beane and manager Bob Melvin addressed a wide variety of issues Tuesday, including the makeup of the infield after re-acquiring Jed Lowrie last week. “We’ve got a few more moves potentially before we comment on who’s where, but I do like the versatility of the guys we have,” Melvin said.

Incumbent shortstop Marcus Semien led the majors in errors, but Beane noted that Semien had a much better second half defensively. “There are some good things about Marcus’ game, and certainly the last month, the trend line was good,” Beane said.

Beane refuted any notion that the A’s are rebuilding. They’ve said that they would like to hang onto their younger players rather than dealing them as they’ve had to in recent years to shore up their pitching staff, but that doesn’t equal a rebuild.

“We’re always trying to win now,” Beane said. “This market isn’t going to allow us to punt for five years, and we’ve never done that.”

Beane reiterated that the A’s see Jarrod Parker, who is coming off his second Tommy John surgery and a fractured elbow, as a starter unless there is medical evidence that he’d have to be a reliever.

Susan Slusser is a San Francisco Chronicle staff writer. E-mail: sslusser@sfchronicle.com Twitter: @susanslusser