Reports have surfaced that Washington may be end up being the only team to offer Lee a seven-year deal. A source told MLB.com's Bill Ladson that "the Nationals are not out of it by any means."
Nationals general manager Mike Rizzo acknowledged on Monday that he had discussion with Braunecker.
Braunecker met with the Rangers on Tuesday and said he expects to meet with them and other teams before the Meetings end on Thursday.
"There is nothing that prohibits us from meeting as many times as we need," Braunecker said.
Asked about the Rangers' interest, Braunecker said, "Sincere."
Asked about their discussions, Braunecker said, "Positive, productive."
He also said "same thing" when asked about his discussions with the Yankees.
Braunecker is also expected to meet with the Yankees again before the Meetings are over. The Yankees, like the Rangers, have made it clear that Lee is their top priority now that they have completed deals with shortstop Derek Jeter and closer Mariano Rivera.
"While we're in the same area, I'm going to meet with them as much as I possibly can," Yankees general manager Brian Cashman said.
Lee, who was 12-9 with a 3.18 ERA in 28 starts for the Mariners and Rangers last season, is the top free-agent pitcher on the market and could command a five-to-seven-year deal worth in excess of $20 million annually.
The length of the contract could be the deciding factor. The Yankees are believed to be willing to go six years on Lee, while early indications are the Rangers would prefer to go no more than five. But that could change as the discussions continue here at the Winter Meetings.
It also remains possible that these Meetings could end on Thursday with Lee still undecided where he will pitch next year.
"I don't know how to describe it," Braunecker said. "We continue to progress. We still don't have any timetable on this."
The Rangers continue to explore other options, most notably talking to the Royals about Zack Greinke. Those discussions appear to be going nowhere.
Depending on which side is talking, the Royals are either asking too much for Greinke or the Rangers aren't offering enough. Either way, nothing appears imminent.
The Rangers have also touched base with other free-agent pitchers, including Carl Pavano, but club officials have made it clear that much still depends on what happens with Lee.