"I think it's pretty cool," A's manager Bob Melvin said. "I talked to him, and he's really excited. I think everybody is pretty excited about that, not just us on the team, but the fanbase might like that one as well.
"I know Klay a little bit, and I'm sure he's pretty happy about this, too. This makes some sense, and man, the times that we've seen him this spring, whether it was with the White Sox or Dodgers, he's swung the bat well."
Thompson will join the A's in Anaheim on Friday for the start of a three-game series against the Angels, per Melvin, but it's unclear just how long he'll be able to stick around.
The A's will soon enjoy an 11-day stretch that includes three off-days, which will allow them to accommodate Thompson while deploying a four-man rotation. But when they need a fifth starter on April 17, Thompson could potentially be the odd man out.
Melvin wasn't ready to speculate on his role for the time being.
"I'm not sure at this point," he said. "We know we will see him in Anaheim. He will be on the team then. We'll see where we go with it."
Thompson hit .255 with a home run -- against the A's, no less -- in 20 games with the Dodgers this spring, before he was designated for assignment. He was claimed off waivers by the Yankees on Tuesday.
The 27-year-old enjoyed three stints with the Dodgers in 2017 but spent the bulk of the season with Triple-A Oklahoma City, hitting .212 with nine home runs and 33 RBIs in 95 games. He has 19 home runs in 151 career big league games, his first 44 coming with the White Sox.
To clear a spot for him on the 40-man roster, the A's transferred right-hander Paul Blackburn (right forearm strain) to the 60-day disabled list.
Jane Lee has covered the A's for MLB.com since 2010.