Ken Rosenthal of Fox Sports reported the two sides were not close on a deal.
Reed, 28, has become a hot trade commodity as teams look for late-innings help, as the Mets' nearly nonexistent playoff hopes could make them more willing to part with their stand-in closer in exchange for prospects.
The Mets also acquired AJ Ramos from the Miami Marlins on Friday, ostensibly ensuring Reed would be headed elsewhere before the deadline.
"I've been traded twice before. If it's going to happen, it's going to happen," Reed told reporters Tuesday. "Nothing I can do about it. I am going to come to the field tomorrow, and if my nameplate is up here, I am going to suit up and go. If not, I'll go some place else."
The Red Sox are attempting to improve a bullpen that already ranks third in baseball in relievers' ERA. Adding Reed as a setup man for Craig Kimbrel would perhaps give Boston MLB's best relief corps heading into what is sure to be a heated stretch run in the American League East.