Dodgers are finalizing deals with relievers Chris Perez and Jamey Wright

In a matter of weeks, the Dodgers’ bullpen went from half-empty to arguably the most stacked in the recent history of the franchise.

The Dodgers were in the process of finalizing one-year contracts Monday with right-handers Chris Perez and Jamey Wright, according to people familiar with the situation who spoke under the condition of anonymity because the deals haven’t been officially announced. This month, the Dodgers added Brian Wilson and J.P. Howell.

With Kenley Jansen entrenched as the team’s closer, Perez figures to be one of three former All-Star closers pitching in non-closing situations. Wilson and Brandon League will be the others.

Wilson is projected as the eighth-inning pitcher, with the 28-year-old Perez a candidate to pitch in the seventh inning. League pitched poorly for the Dodgers this year but was clocked regularly in the mid-90s, making it conceivable he can recover the form that landed him a three-year, $22.5-million contract.

Perez had a career-worst 4.33 earned-run average this year with the Cleveland Indians, but that was largely a reflection of an abysmal final month. Perez’s ERA was a respectable 3.15 through Sept. 1 and the two-time All-Star finished the season having converted 25 of 30 save opportunities.

The Indians released Perez on Oct. 31 rather than pay him the $9 million or so he was expected to earn in the arbitration process.

Perez will draw a base salary of $2.3 million from the Dodgers, according to a person familiar with his contract. Performance-based incentives can increase the value of the contract to somewhere around $6 million to $8 million.

The financial details of Wright’s contract aren’t known.

Wright, who turns 39 Tuesday, pitched for the Dodgers in 2012. He had a 3.72 ERA in 66 games that season. He moved this year to the Tampa Bay Rays, with whom he had a 3.09 ERA in 66 games.

Wright figures to be the Dodgers’ long reliever.

Howell, who agreed last week to a two-year, $11.25-million contract, is expected to be the primary left-handed option in the late innings.

Paco Rodriguez will be looking to recover from a September burnout to be the second left-handed reliever. But Rodriguez could face competition for the final spot in the bullpen from several hard throwers, among them Chris Withrow, Jose Dominguez, Rule 5 draftee Seth Rosin and onetime closer Javy Guerra.

Withrow, 24, was particularly impressive in his rookie season this year, recording a 2.60 ERA in 26 appearances.

Left-hander Scott Elbert could return from reconstructive elbow surgery in the middle of the season.

dylan.hernandez@latimes.com

Twitter: @dylanohernandez