"Yeah, I do," Dombrowski said when asked if he has any idea when it will end, "but I'm not really ready to say that, because I don't want you guys calling me every hour or every three minutes. I have an idea."
Dombrowski cautioned, though, that no deal is imminent. Damon would have to undergo a physical before completing any deal.
"We're not at that juncture," Dombrowski said. "I don't think anybody else is."
Details of a Tigers offer first emerged in reports last week. Multiple reports have suggested the Tigers have offered a two-year, $14 million contract. The Lakeland Ledger cited a source who would be very surprised if the Tigers offered anything more than a one-year contract.
Damon also has drawn interest from the White Sox and Braves, among other clubs. White Sox GM Ken Williams told the Daily Herald that he would talk about Damon on Sunday.
When asked why the Tigers would be interested in Damon, Dombrowski echoed some of the sentiments that have been mentioned about Damon by agent Scott Boras and others.
"First of all, he's a good player," Dombrowski said. "He would help us from an offensive perspective. He's another left-handed bat. He's been a winner. So there's a lot of things about him that are very good in that regard. He's a good player.
"It's one of those things where, he being out there at this time of the year, you normally wouldn't anticipate that. So we've made some adjustments ourselves. Of course, anytime you do that at this time of year, [owner] Mike Ilitch was involved in that. And so, we just decided that we would go ahead and try to make an offer, because we think he would help us."
Dombrowski would not go into detail on many issues surrounding the interest, including the level of involvement from Ilitch, details on the contract, or whether it comes with a time limit.
"I would assume, like any player, that they want to get into camp and get ready for the season," Dombrowski said. "So at some point, there's a timeframe there. But I think he knows that."
Dombrowski also declined to get into the Tigers' financial dealings at this point, including putting the Damon offer in the context of the Curtis Granderson trade in December. Those are issues he said he would address once Damon signs somewhere.
As to his mood on their chances of making a deal, he was nondescript about that, too.
"I'm not closing it off," he said, "but I'm not saying it's done by any means. We're still in it."
To Dombrowski's knowledge, nobody from the Tigers has contacted Damon in any sort of recruiting effort. When asked about reports that White Sox catcher A.J. Pierzynski and broadcaster Ken Harrelson played golf with Damon recently, Dombrowski smiled and joked.
"Maybe that helped us," Dombrowski said with a laugh. "I don't know."