CLEVELAND — With their good young talent proving hard to lock up to long-term deals, the A’s are turning their interest toward designated hitter Khris Davis, who can be a free agent after next season and who repeatedly has said how much he’d like to stay in Oakland.

According to multiple sources, the A’s have had talks with Davis’ agent, Lou Nero of Octagon, about a multi-year deal. It’s unclear, however, if the sides will be able to hammer out a deal that extends beyond 2019 — adding one year onto the deal doesn’t really sweeten the pot much for Davis, who would get a minimum one-year deal as a free agent, anyway.

To get Davis signed to a multi-year commitment, it’s likely Oakland would have to go at least three years, and when it comes to their own players, the A’s mostly have veered away from deals of more than two years. Reliever Sean Doolittle, who signed a five-year extension at very favorable terms for the team in 2014, is the most recent to get a longer deal with the A’s, and in 2008, second baseman Mark Ellis got two years plus a team option — potentially a model for talks with Davis.

The free-agent market for power-hitting DH types has not been a rich one the past two seasons, but Davis, 30, is aiming for a third consecutive 40-home run season.

“I’m keeping an open mind,” Davis said. “I don’t know too much about this stuff, that’s why I have an agent. I just play. I hope I bring the best with whatever kind of deal I get. The performance is my expertise.”

The A’s would like to lock up Matt Chapman, 25, and Matt Olson, 24, but their representatives are leery about buying out their arbitration years. After last year’s slow offseason, many agents believe clients should be as young as possible when they hit free agency.

Briefly: The A’s will decide Saturday whether left-hander Brett Anderson (shoulder) will come off the disabled list Sunday or Monday or make one more rehab start in Triple-A. They’ll also have a better idea then when right-handers Daniel Mengden (foot) and Trevor Cahill (Achilles tendon) will be ready to go. Mengden allowed two hits and one unearned run in 32/3 innings for Triple-A Nashville on Friday night. He struck out five and walked none. ... Double-A Midland starter Jesus Luzardo and catcher Sean Murphywill play in the July 15 Futures Game at Nationals Park. ... The A’s did not sign any draft picks at the deadline Friday. They came to terms with 35 of the 40, including their top 24 picks. Florida catcher J.J. Schwarz (eighth round) and Washington starter Joe DeMers, an 11th-rounder from Pleasant Hill, signed in the past two weeks.

Susan Slusser is a San Francisco Chronicle staff writer.