"I don't want to say what our offer was, but we made what we considered to be the best offer we could do," Melvin said. "They have to make the determination about the gap, if there's a big gap in money. And there might have been. The other thing is, there might have been other teams involved, too."
Dempster has spent all of his 15-season career in the National League aside from 12 starts with the Rangers to end last season. He threw 173 innings last year between Texas and the Cubs, and in the four previous seasons, he reached 200 innings.
The Sox notably have lacked durable arms.
"We struggled in that area for different reasons," Cherington said. "One of the things we've been lacking has just been the reliability and someone who can be a reliable, durable part of the rotation. So that is something we've focused on this offseason. ... I think we feel like need to go into 2013 with more starting pitching depth than we have right now. There are different ways to do that."
The Red Sox might feel this is the time to push for pitching, with moves expected to come faster ahead of Christmas and the new year.
"There's this period between now and 12 days from now, stuff tends to get done this time a year, then maybe there's a quiet period and things pick up again," Cherington said. "There are opportunities to get things done in the next 12 days, and after that, we'll keep working on it."
The largest question surrounding Dempster is the adjustment not just to the American League, but the AL East. Dempster's time in Texas didn't go as planned. After posting a 2.25 ERA with the Cubs, he had a 5.09 mark with the Rangers. That included 10 home runs in 69 innings: Dempster allowed one fewer home run (nine) in a lot more innings with the Cubs (104).
But pitchers have been able to transition from the NL to the AL. Sometimes there's an element of surprise that works against hitters, but that's not likely to mean much across a whole season.
"Harder. It's certainly harder," Cherington said of the switch across league lines. "When you talk about [the Yankees'] Hiroki Kuroda, it's easier because you just saw it in the AL East. You don't have that kind of recent history with everyone. That doesn't mean we can't find the right guy. Just got to keep working at it."
Adding Dempster probably doesn't mean the Sox are content with their stockpile of arms. Cherington said he's been involved with nearly every starting pitcher that's come off the market, be it by trade or free agency. That list, then, could include Zack Greinke and Dan Haren.
"I think we've been involved at some level in just about every starting pitcher that's moved -- either signed or traded," Cherington said. "Not everyone, but just about everyone. In every case, you sort of have a limit, boundaries for what you're willing to do for each situation, and we haven't been able to execute it yet.
"We're still hopeful we'll be able to add somebody or more than one guy. We're not going to stop necessarily. We'll see. I don't look at [those pitchers who have new teams] as missed opportunities, because you pursue things, but you have to have a limit of what you're going to do. We've had a limit, and at some point, we'll find a match. I'm hopeful and confident we'll be able to do that."
Dempster had a 43.5 percent ground-ball rate, 51st best among Major League starters, according to FanGraphs. His fastball appears to have dropped into the 89-mph range on average, whereas he sat in the 90-mph range the previous three seasons, according to the site.
FOX Sports first reported that the Sox reached agreement with Dempster.