On Wednesday, Bob Nightengale of USA Today passed along word of the talks but noted they stalled because Boston didn't want to include outfielder Jackie Bradley Jr.
The Red Sox could use a power upgrade after finishing 27th in home runs during the 2017 season.
They have been consistently linked to free-agent outfielder J.D. Martinez, who cranked out 45 homers for the Arizona Diamondbacks and Detroit Tigers last year, but they haven't come to an agreement. Ken Rosenthal of The Athletic reported Tuesday the slugger is "fed up with the Red Sox's inflexibility" in contract talks.
Encarnacion could be another way to bolster their HR output.
The 35-year-old Dominican Republic native has tallied at least 34 home runs in six straight years, dating back to 2012 with the Toronto Blue Jays. That includes 38 long balls last season, his first with the Tribe after signing a three-year, $60 million deal.
Moving Bradley to acquire Encarnacion would create some lineup issues for Boston, though.
Mookie Betts would likely be forced back to center field to fill the void, leaving new manager Alex Cora to decide whether to play Hanley Ramirez or Mitch Moreland in a corner outfield spot so everybody could fit in the lineup with Encarnacion playing first base or designated hitter.
Alas, the offensive boost the team would receive in that scenario would probably get offset by the defensive weakness it creates.
While the fit isn't ideal, Nightengale reported it's possible the trade talks pick back up depending on whether the Red Sox eventually land Martinez.