Would the Boston Red Sox trade right-hander Josh Beckett? Of course they would.

Will such a deal happen before Tuesday’s non-waiver deadline at 4 p.m. ET? Probably not.

The Red Sox discussed Beckett with a “small handful” of teams that inquired about the club’s starting pitchers, but a deal is unlikely, according to a major-league source.

An executive from a team in need of pitching said the Red Sox asked his club, “Would you?” about Beckett. That, however, was the extent of the conversation.

Beckett, 32, is 5-9 with a 4.57 ERA this season, and some in the industry believe that the Red Sox would improve their culture by trading him – he was part of the infamous fried-chicken-and-beer episode at the end of last season.

However, Beckett is under contract through 2014 at salaries of $15.75 million per season. He also has the right to veto any trade as a player with 10 years of major-league service, the last five with the same club.

While Beckett has said he would consider waiving his no-trade rights for the right team, his mediocre performance and remaining salary would make it difficult for the Red Sox to get a significant return.

Jeff Passan of Yahoo Sports reported that the Red Sox reached out to the Texas Rangers and Atlanta Braves about Beckett. Jon Heyman of CBSSports.com was the first to report that the Sox were contacting teams about him.

-Ken Rosenthal