Cardinals general manager John Mozeliak said he expects to hear back from Holliday's agent, Scott Boras, within 48 hours. Terms of the deal are unknown.
"Frankly, the importance of winter meetings was to get a better understanding of what other players are looking for. To present an offer to Matt is important," Mozeliak told the St. Louis Post-Dispatch. "Now we need to hear back from them on where this may go. We hope there is some dialogue and hopefully use this weekend to better learn where we stand."
Boras touched on Holliday's situation from several angles during a 35-minute, give-and-take with reporters Wednesday. He compared Holliday's production to Mark Teixeira's (but declined to say if that equates to a $180 million, Teixeira-type contract).
A source told the Post-Dispatch the Cardinals' offer did not average $18 million annually -- the number that Holliday refused to accept when he was with the Colorado Rockies in 2008.
Boras suggested the Cardinals could make it more enticing for Albert Pujols to stay in St. Louis by keeping Holliday and strengthening their lineup. And he stressed that Holliday's suitors have just as much input in the timetable as he does.
"The time frame of these things is predicated more on the clubs," Boras said. "I don't write checks. I can go in and talk about a lot of things, but I don't make that decision."
Holliday, 29, is a three-time All-Star and the 2007 National League MVP runner-up.
Information from ESPN.com's senior baseball writer Jerry Crasnick was used in this story.