Jesse Chavez, valuable in every role possible for a pitcher except closer the past three seasons, was dealt to the Blue Jays on Friday for reliever Liam Hendriks.

Chavez, 32, is just a year removed from free agency and the A’s are trying to rebuild their bullpen. Hendriks, 26, has four years of club control remaining, and the Australian is coming off a breakthrough season in which he pitched exclusively in relief for the first time and went 5-0 with a 2.92 ERA, with his fastball jumping from the low 90s to mid-90s.

“He took a huge leap moving to the bullpen, velocity, performance,” A’s general manager David Forst said by phone.

The A’s coaching staff and front office always has appreciated Chavez enormously and, Forst said Friday, “He busted his butt for us for three years, but this trade made sense for us.”

Forst said that the A’s view Hendriks as a back-end of the bullpen pitcher, which means he’s likely to pitch in a set-up role.

This move will not affect Oakland’s pursuit of a free-agent reliever and as The Chronicle reported earlier Friday, the team has interest in Darren O’Day, the top available reliever.

Forst said that he and Chavez had a great talk Friday and Chavez said of the A’s, “This is probably not the last time our paths cross.

“This is the first team I’ve been with for a long time, so this is all hard to fathom,” Chavez said. “But I know the Blue Jays organization and I know what a great team team they have and that’s the best way to get over the initial shock, that’s something to get excited about.”

Chavez made nine appearances with the Blue Jays in 2012 before the A’s purchased him; he’d played for seven organizations by the time he joined the A’s, and he said with a laugh Friday, “The Blue Jays are the first organization that’s wanted me back of the seven and that’s a big plus - I’m glad I made a good impression.”

Chavez went 17-27 with a 3.98 ERA with Oakland over three-plus years, but he often had little run support - and he had some spectacular early season runs, including a 1.93 ERA in six starts last May and a 1.89 ERA in six starts in April, 2014. The second half has been when the slight right-hander has struggled, his ERA going up more than 1.50 in each of the past two seasons after the All-Star break.

Immensely popular in the clubhouse, Chavez was known for helping other pitchers spot delivery flaws and suggesting new pitches and grips; many of them believe he will be an outstanding pitching coach once his career is over.

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