The deal is worth $7.5 million, sources told ESPNDallas.com.
Pierzynski, who recently had lunch with Washington and other Rangers officials, said he is excited to play for Washington and isn't upset anymore.
"It came up (at the lunch), and I made a joke of it," Pierzynski said on a conference call with reporters on Wednesday. "It was over as soon as it happened. You're disappointed as a player because you want to go to the All-Star Game and be honored that way. But when it was over, it was over."
Pierzynski said he's talked to some players who have played for Washington and has heard "nothing but praise."
Washington managed the American League All-Star team for the second straight season in 2012 but didn't have room on the roster for Pierzynski. He said at the time that it was the toughest omission.
"I know he was in a tough spot," Pierzynski said. "It wouldn't change how I feel about Ron Washington because I know what he's done in this game and where he stands. I'm over it and hopefully we can move on from that also."
Washington said Wednesday the whole episode is "old news" and that he's moving forward, saying he's pleased to have Pierzynski on the Rangers.
"I always believed he did anything and everything he had to do to try to beat you," Washington said.
Pierzynski was asked a series of questions about his reputation in the game, including what he thinks Rangers fans should know about him.
"The only thing they need to know about me as a person is that I want to win," Pierzynski said. "Whatever is said or been written, you can't change that. You can't make people change what they've said or what they've done. All I can do is play as hard as I can."
Pierzynski, named Texas' starting catcher by Washington, said he wants to look himself in the mirror and say that he "gave it all I could for the Texas Rangers in 2013."
Pierzynski fills Texas' hole at catcher, joining Geovany Soto as the Rangers' two primary backstops for the 2013 season. Both players have short-term contracts; Soto signed a one-year deal worth $2.5 million earlier this offseason.
Pierzynski, who turns 36 on Sunday, batted .278 with 27 home runs (a career high) and 77 RBIs (tied a career high in 2004) in 135 games this past season for the Chicago White Sox.
"He's a winning guy," Rangers general manager Jon Daniels said. "There just aren't that many catchers in the league that go out there and play every day. It is a demanding position and that says a lot about him being ready to go."
Pierzynski gives the Rangers a left-handed power bat in a lineup looking to fill voids left by Josh Hamilton, who signed a five-year contract with the Los Angeles Angels, and Mike Napoli, who agreed to terms on a three-year deal with the Boston Red Sox. That deal has not been finalized.
Napoli hit 24 homers and had 56 RBIs in 108 games for Texas in 2012, but he hit just .227.
In 133 career at-bats at Rangers Ballpark in Arlington, Pierzynski has hit .271 with 3 homers, 16 RBIs, 9 doubles, 1 triple, 19 strikeouts and 6 walks.
"It's been a long time since I changed teams," said Pierzynski, who has played the past eight seasons with the White Sox. "I'm looking forward to the first day of spring training and even the Fan Fest (in January in Arlington) to get to know some of these guys. I can't wait to know them a lot better.
"To walk into the clubhouse for the first time and meet new people, I'm looking forward to it. It definitely puts a pep in your step and changes things."