According to Jon Morosi and Ken Rosenthal, the Royals are "casting a wide net" in the trade market, and are hoping to upgrade their team offensively as well as on the mound.
As Rosenthal states, Ben Zobrist makes sense for almost every team looking to make a push for the playoffs, as his versatility allows him to play every infield spot, as well as all three outfield positions. He's one of the few valuable second baseman on the market in 2015, and will likely have many teams bidding for his services. Zobrist is almost guaranteed to be traded, however as Rosenthal points out, they're in no hurry to start breaking up their roster as they try and get back into the race before it's too late.
While the Royals don't have one of the best farm systems in the league, Baseball America has them ranked 13th overall, and could realistically pull off a trade for Johnny Cueto. He's not going to be cheap, but the fact that he's a rental player will help bring down the asking price to where the Royals won't have to mortgage their future to acquire him. The Tigers, Yankees, and Giants had been mentioned previously as potential landing spots for the right-hander, and while they still might be in the hunt, Morosi writes that the Reds are still reluctant to engage in serious trade talks, and are likely to wait until after the All-Star break.
Kansas City has also been linked to Scott Kazmir, who's almost certainly going to be traded if the A's can't get back into the playoff race before the middle of July. He's a free agent at the end of the year, and one of the things we know about Billy Beane is that he hates to watch players leave without getting some value back. If he ultimately deems this season lost, Oakland is going to sell off a lot of pieces, and begin restocking for the 2016 season.
The Royals were not picked by many to be competitve this season, but with June coming to a close, and Kansas City still in first place, they're primed to be very active in trade talks before the deadline passes. Their rotation ranks 25th in terms of fWAR, and is clearly in need of some help. They've had to deal with injuries throughout the season, but despite a subpar core of starters, the Royals are still 13 games over .500. If they can acquire an ace like Cueto, or pieces like Kazmir and Mike Leake, they could be set up incredibly well for another deep October run.