GLENDALE, Ariz. -- The White Sox added free-agent right-handers Joe Kelly and Vince Velasquez to their deep and what they hope is a championship-level bullpen on Monday.

Kelly, 32, agreed to terms on a two-year, $17 million contract. He will receive $7 million in 2022 and $9 million in '23, with the White Sox holding a $9.5-million option for '24 with a $1-million buyout. Velasquez, 29, agreed to terms on a one-year, $3 million contract.

The White Sox on Tuesday also announced the signing of free-agent second baseman Josh Harrison, who inked a one-year, $5.5 million deal. Harrison's contract also includes a $5.5 million option for 2023 with a $1.5 million buyout.

Kelly won’t break camp with the White Sox to open the 2022 season, as announced by general manager Rick Hahn. He sustained a right biceps nerve injury while making his seventh and final 2021 playoff appearance for the Dodgers during Game 5 of the National League Championship Series, and the White Sox will take the proper time to build him back up, especially when considering this shortened Spring Training following the lockout.

“If we had a normal spring, maybe he’d break with us,” Hahn said. “We’ll have to see how the next several weeks of his build back goes. There’s just going to be no rush here. Could be a few weeks into the season.

“We’ll just have to wait to see until he starts facing hitters and building him back. Nothing gives us long-term pause in this. This is about making sure a guy, who obviously has not been under a team’s care for three-plus months and now is showing up for an abbreviated spring, isn’t rushed.”

In 48 appearances last season for the Dodgers, Kelly posted a 2.86 ERA and a 27 percent strikeout rate. He owns a career 3.83 ERA, also pitching for the Cardinals (2012-14) and the Red Sox (2014-18). The hard-throwing right-hander has extensive postseason experience, appearing in 40 games with a 3.55 ERA over 58 1/3 innings, winning World Series rings with Boston (2018) and Los Angeles (2020).

A connection exists between Kelly and White Sox manager Tony La Russa from his Cardinals time, when La Russa was manager, and during his Red Sox stint, when La Russa was part of the front office.

“Obviously everyone knows Tony,” Kelly said. “Tony is a little bit more old school, which is kind of what I like. I kind of like, if you suck, you suck type of mentality. I don’t need a pat on the back when I’m doing bad. Just tell me to do better.”

“His first big league camp was 2011, and I can remember to this day -- we were doing the same thing we're doing now,” La Russa said of Kelly. “We didn't have a whole bunch of batting practices at one time, you had one at a time so you could see everybody, and I was saying this to [Dave] Duncan, 'Wow, man.' We didn't know about him. And we found out they were going to send him to the Florida State League, and I said, ‘I want to manage in the Florida State League.’ He was that good looking.”

Kelly was “letting it rip” off flat ground in the outfield during Monday workouts, being cautioned to stop throwing all three pitches. So, he doesn’t seem worried about the delay to his ’22 season.

“It’s smart to play it safe, not just for myself, but with every single guy we have in this clubhouse because when you lose one of your guys the morale of everyone else is going to be [affected] a little,” Kelly said. “As long as we play it safe and all across the board with our young guys, old guys, I think it’s the smarter move, for sure.

“You can't really do much for it. Just let it heal and I've been throwing since the beginning of February, right before February. Throwing is not the issue. It’s just not doing anything crazy. I'm definitely excited. I want to be throwing as much as I can and be ready to face Eloy [Jiménez] and José [Abreu] and all those guys. My time will come and I'm super-excited to be there.”

Velasquez has appeared in 156 games (127 starts) over seven big league seasons. He spent his entire career with the Phillies before being released last September and signing with the Padres for the final month of the season. The right-hander has posted a 4.95 ERA with 716 strikeouts over 651 innings.

There’s a Spring Training plan to stretch out Velasquez, along with Reynaldo López and the rest of the rotation, giving the White Sox length on their staff. Hahn pointed to five innings as the best-case scenario for any starter in a Cactus League game, with the schedule beginning on Thursday, so there’s a need for reinforcements.

“Having as many quality arms that can give us multiple innings is a good thing to have,” Hahn said.

Liam Hendriks, who was the 2021 American League Reliever of the Year, returns as the closer coming off a 38-save season. He is joined now by Kelly, fellow free-agent addition Kendall Graveman and left-handers Aaron Bummer and Garrett Crochet. This bullpen for the reigning AL Central champs also includes Craig Kimbrel, who enters the 2022 season as the active saves leader with 372.

Kimbrel was acquired from the Cubs at the Trade Deadline last season for second baseman Nick Madrigal and reliever Codi Heuer in the hopes of putting together a late-inning force. But Kimbrel, who was the National League’s top closer in the first half of the 2021 season, struggled with a 5.09 ERA over 24 games in a non-closer’s role for the White Sox.

Trading Kimbrel was possible even before the Kelly addition. But Hahn certainly won’t give away a potential valuable contributor, and one on whom the White Sox exercised a $16 million option for the 2022 season.

Harrison, 34, is a versatile infielder and outfielder who spent last season with the Nationals and A’s, hitting .279/.341/.400 in 138 games (90 with Washington and 48 with Oakland). The two-time All-Star has a career .719 OPS in 11 Major League seasons and would fill Chicago’s vacancy at second base. Harrison also gives the White Sox a chance to use versatile players such as Leury García, Danny Mendick, Romy Gonzalez and Yolbert Sanchez around the diamond.