It appeared that pragmatism might trump precedent when sources confirmed to Newsday on Tuesday that the Yankees checked in with the Mets about a rare crosstown swap that would send reliever Addison Reed and first baseman Lucas Duda from Queens to the Bronx.

But that possibility became a lot more remote late Tuesday night when the White Sox announced they had traded relievers David Robertson and Tommy Kahnle and third baseman Todd Frazier to the Yankees for reliever Tyler Clippard and three prospects.

So Reed no longer is a need for the Yankees, while adding Duda became far less of a priority. The clubs have not made a player-for-player trade since 2004, when the Yankees sent Felix Heredia to the Mets for Mike Stanton in a swap of relievers.

Even before word of the White Sox trade, the Yankees appeared unlikely to part with top-tier prospects for Duda and Reed. The Mets appeared better positioned to receive a stronger return for Reed from other teams, with the Yankees envisioning him as a seventh-inning type who would bolster the bullpen behind Dellin Betances and Aroldis Chapman.

Either way, Reed, 28, remains the Mets’ most valuable trade chip, and he has received healthy interest from other clubs leading up to the deadline. The righthander has 15 saves and a 2.47 ERA in 42 games this season. He will be a free agent this offseason.

Duda, 31, could have given the Yankees a fix for their problems at first base, with Greg Bird having undergone what could be season-ending ankle surgery. The power-hitting lefty has a swing that could be ideal for Yankee Stadium. In 67 games, he has 16 homers and 34 RBIs.

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The Yankees and Mets have shared New York City since 1962, but the franchises have made only 16 trades with one another, the last coming in 2014, when righthander Gonzalez Germen was sent to the Yankees for cash.

Whether a trade actually gets made, the brief discussions only further highlight the divergent paths the crosstown rivals have traveled since spring training. Although the Yankees have lost 21 of their last 31 games, they remain in the mix for a playoff spot in the American League, thanks to a fast start powered partly by the unexpected emergence of Aaron Judge. The Mets began the day 10 1⁄2 games out of a wild-card berth, far short of the lofty expectations they carried into the season.

General manager Sandy Alderson has been public about his desire to move veterans on expiring contracts leading up to the nonwaiver trade deadline July 31. With the Mets already essentially out of contention, sources said yesterday that players have been notified that playing time could be altered to further showcase those who could be moved. In addition to Reed and Duda, the Mets are open to trading outfielders Jay Bruce and Curtis Granderson and infielders Asdrubal Cabrera and Neil Walker.

The task of unloading those players comes with complications. Though Cabrera publicly declared he would be open to a trade, a source said the Mets have received minimal interest. A report in the Boston Herald labeled Cabrera as a potential target for the Red Sox, who need a third baseman. But a source said yesterday that Boston has yet to ask the Mets about Cabrera.

Moving Bruce for a worthwhile return also could be a challenge. Yesterday, the Tigers traded J.D. Martinez to the Diamondbacks for three prospects. It was considered a light return by some rival executives, which could impact what Bruce could fetch in the trade market.

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Bruce, 30, is hitting .267 with 24 homers and 62 RBIs. But sources told Newsday last month that officials have not ruled out a scenario in which Bruce is retained and given a qualifying offer as part of an effort to keep him long term.