That leaves right-handers Jeff Niemann and Roberto Hernandez as the finalists for the spot.
Archer, the Rays' No. 4 prospect, pitched particularly well this spring. The 23-year-old right-hander allowed no runs on one hit and three walks while striking out five in three outings.
"I'd be lying if I said I wasn't [surprised], just because I've shown up pretty well in camp," Archer said. "But it was a great conversation with Joe [Maddon], Andrew [Friedman] and [Jim] Hickey. And they just kind of laid it out for me and told me that I needed to go down to work on some things, continue to get better, continue to build off the second half of my Minor League season last year. And after the conversation, I was at ease, you know. There was no more surprise. I was pretty happy how the conversation went."
Archer is pleased with the way he has pitched.
"And they noticed what I was doing, working on my changeup, working on some things in different situations," Archer said. "They pointed that out, and they liked the fact that I came in working to earn a spot, but I came in to work on certain pitches and certain things. It was a great conversation."
Archer went 1-3 with a 4.60 ERA in six games with the Rays last season, and four of those appearances were starts.
"Mixed with my conviction that I can compete at this level and with that conversation that we just had, there's no like malicious feelings going to Triple-A, going to the Minor Leagues for who knows how long," Archer said. "I know I have to continue to develop.
"Something's going to happen every year. Somebody goes down or they need an extra arm or whatever, so I'm going to make the most of my opportunity when I come up, but continue to maximize my potential in the Minor Leagues."
Odorizzi was one of the players who came over to the Rays in the trade that sent James Shields and Wade Davis to the Royals.
"Both of them, we feel, are a big part of our future," said Maddon of Archer and Odorizzi, Tampa Bay's No. 3 prospect. "We wanted to get them down pitching on a regular basis. For sure, we believe Archer could be here right now. But you can't have everybody, so we want to keep him ready in the event something were to happen, he'd be stretched out properly and be able to jump right into it. It's kind of a nice problem to have when you're that thick that you're sending back guys who are that good."