Hillman was about to leave Japan after saying goodbye and being feted for his five years as manager of the Nippon Ham Fighters.
General manager Dayton Moore did not discount the Royals' interest in Kuroda.
"We're going to explore every opportunity available to us," Moore said. "He's made his wishes known that he'd like to come to the United States. But it's not in the best interests of the Kansas City Royals to tell the other 29 clubs what we're going to do."
The Royals already have a tentative agreement with Japanese reliever Yasuhiko Yabuta, a deal that could be finalized Wednesday if he passes a physical examination.
However, a starting pitcher to go with Gil Meche, Brian Bannister and Zack Greinke is also on the Royals' wish list.
There's a line forming, though, for Kuroda, who has a 103-89 record and a 3.69 ERA in 11 seasons for the Hiroshima Carp.
The Seattle Mariners are considered to have the inside track. They sent general manager Bill Bavasi and manager John McLaren to Japan recently to recruit Kuroda. In the Mariners' favor: Kuroda prefers playing for a West Coast team and the club already has Japanese stars in outfielder Ichiro Suzuki and catcher Kenji Johjima.
The Arizona Diamondbacks made an offer to Kuroda about 10 days ago. General manager Josh Byrnes said he's waiting for word from Kuroda's agent, but speculated that the wait might extend into next week's Winter Meetings.
Other clubs believed to be interested include the Chicago Cubs, Los Angeles Dodgers, Philadelphia Phillies and Texas Rangers.
The speculation is that Kuroda, 32, could command $10 million a season in a multiyear contract.
"I've been told from reliable sources that the process with Kuroda and where he's going is likely to take a little time," Hillman said. "But he'd help us get stronger immediately."
Hillman had hoped to visit with Kuroda during his six-day trip to Japan, but it didn't work out.
Last season Kuroda had a 12-8 record and a 3.56 ERA in 26 starts with 123 strikeouts and 42 walks in 179 2/3 innings. In 2006, he had a league-best 1.85 ERA and in 2005 he had a league-high 15 victories.
Kuroda comes armed with a mid-90s fastball that Hillman labels as "above average," plus a slider and a split-finger.
Hillman remembered that after Kuroda beat his Fighters a couple of years ago, he asked the Carp for permission to talk to the pitcher the next day.
"That's the only time in my five years here that I asked to meet with an opposing Japanese pitcher," Hillman said. "But I told him what a great competitor he was. He just dominated us. I told him how impressed I was with him."
Kuroda signed a four-year, $10.5 million extension with the Carp before last season, with a provision that he could become a free agent after 2007, an option he exercised.
"Certainly Trey has a comfort level with him," Moore said. "He's seen him pitch the last five years and we'll evaluate every player that has interest in coming to Kansas City. Although I don't know that he does."