BOSTON -- On the eve of Tuesday's 4 p.m. ET non-waiver trade deadline, Red Sox manager Bobby Valentine had plenty of company in his office.

Following a 7-3 win over the Tigers, Valentine entertained team president/CEO Larry Lucchino, general manager Ben Cherington and assistant general managers Mike Hazen and Brian O'Halloran for a lengthy closed-door meeting.

One topic that was likely to have at least been discussed was Josh Beckett. While the Red Sox did talk to some teams -- including the Texas Rangers and Atlanta Braves -- about the possibility of a trade involving Beckett, things had cooled considerably by late Monday, according to a Major League source.

Gordon Edes of ESPNBoston.com, citing a team source, reported definitively late Monday that the Red Sox will not trade Beckett before the non-waiver Deadline.

In fact, Beckett is scheduled to pitch for the Red Sox on Tuesday night against the Tigers.

Becket will earn $15.75 million annually in each of the next two seasons, which was probably a big reason not much of a market has developed for his services.

While the Rangers spent some time on Sunday and the early portion of Monday toying with the idea of making a run at Beckett, talks with the Braves never got off the ground. Atlanta did have interest in Sox lefty Jon Lester. However, they were told that Boston has no interest in moving Lester, who is four years younger than the 32-year-old Beckett.

There are other reasons it would be difficult to move Beckett. He has 10-and-5 veto rights and his value is hardly high right now considering his statistics.

Beckett, 32, is 5-9 with a 4.57 ERA.

As of Friday, Beckett told WEEI.com that the Red Sox had not asked him about waiving his 10-and-5 rights.

Following Monday's win, Clay Buchholz expressed hope that his rotation mate stays put.

"I don't think anyone here wants to see him go," said Buchholz. "He's one of the main pieces of the foundation of this team, I think. His track record shows that he's still one of the best pitchers in baseball. He had a [2.89] ERA last season. He's still the same pitcher these guys signed. He's a guy everyone gets along with in the clubhouse, too. But that's an organizational decision. That's not a player's decision."

The Red Sox, at 52-51, are 8 1/2 games back in the American League East and four games off the pace in the Wild Card standings.

There is a belief within the team -- from the front office to the manager's office to the clubhouse -- that the Red Sox still have a run in them.

After getting pounded by the Yankees on Friday night, Boston showed some fight in winning the last two games of the series in its last at-bats. Then the Sox played one of their best all-around games in a while on Monday against the Tigers.

Valentine has stated numerous times in recent weeks that he'd be fine if Cherington didn't make a major move before the Trade Deadline.

"I made it clear right along -- I wouldn't be disappointed if that happened," Valentine said. "You know, I also went over to Ben's office and I know that he's working as hard as anybody could work to see if there's anything out there that can improve our team even a little."

"Our ownership and organization did a great job of putting together this team in this offseason and years past," said Buchholz. "They wouldn't have done it if they didn't think we were good enough to win. That's where everybody's mindset is at."