But Gardenhire took a light-hearted approach to those trade rumblings with Morneau joining Minnesota natives Joe Mauer and Glen Perkins for the All-Star logo press conference before the series opener against the Royals.
"I asked him how he got in on the 2014 logo if he's not going to be here -- he laughed," Gardenhire said. "You've got to have fun with it. It's serious stuff but still, I want him to be relaxed and play baseball. You can't control that stuff."
Morneau said he's been doing his best not thinking about being traded, and reiterated he wants to remain with the Twins.
"I just have to go out and play," Morneau said. "Hopefully, I'll be wearing the same uniform today, tomorrow and the next day. I'm happy to be here. This is where I want to be. But if there's something that will make this team better in the future, that's up to them. But I want to be part of it and be here."
Morneau, though, did say it's weighing on him more than in recent years, as there's a stronger possibility he's dealt this time around. Teams reportedly interested in Morneau include the Orioles, Pirates, Red Sox and Yankees.
"[This year is different] because I'm so close to the end of my contract," Morneau said. "And with the team not doing as well we'd like, it's always a possibility. I'd like to be here with my family, it's the only team I've ever known in the big leagues, and that's all I can really do."
Twins general manager Terry Ryan wouldn't talk about Morneau specifically, but said he wouldn't trade a player in the last year of a deal just to do it, as he'd still want a quality player in return. He added the Twins are looking for younger players with years of control because they're building for the future, but couldn't predict whether he'd make a move before the Trade Deadline.
"As you can tell it's been a little slow in the industry, but tomorrow it should speed up like it does whenever the Deadline arrives," Ryan said. "Whether we do or do not, I don't have a prediction right now. There haven't been many things happen in the last three or four days, frankly."
Rhett Bollinger is a reporter for MLB.com. Read his blog, Bollinger Beat, and follow him on Twitter @RhettBollinger. Kelly Erickson is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.