Morosi noted that Miller had a strong relationship with Mets manager Mickey Callaway when Callaway was the pitching coach for the Cleveland Indians.
After several dominant years, Miller struggled in 2018 primarily due to injury.
The 33-year-old lefty landed on the disabled list three times last season with hamstring, knee and shoulder ailments.
Miller was limited to 37 regular-season appearances, and he finished with a 2-4 record, earned two saves and posted a 4.24 ERA and 1.38 WHIP. He also struck out 45 batters over 34 innings.
From 2014 to 2017, Miller had an ERA of 2.04 or lower in each season.
Cleveland acquired him in a trade with the New York Yankees in 2016. That season, he was named an All-Star for the first time and went 10-1 with a 1.45 ERA, 0.69 WHIP and 123 strikeouts in 74.1 innings pitched.
The Indians leaned heavily on Miller during their run to the World Series in 2016, as he threw 19.1 innings over 10 playoff appearances, finishing with a 1.40 ERA and 30 strikeouts.
Miller was named the American League Championship Series MVP, but Cleveland fell just short of a World Series title, losing to the Chicago Cubs in seven games.
There was no World Series hangover for Miller, as he followed up that performance with a 1.44 ERA and another All-Star nod in 2017.
While legitimate questions surround Miller's health after last season, he could be a great fit in New York for the right price given his previous success with the crosstown rival Yankees.
The Mets are in need of a closer or setup man after trading Jeurys Familia to the Oakland Athletics last season.
Robert Gsellman, Seth Lugo and Anthony Swarzak are among the top internal options to close, but they both have limited experience in that role.
If the Mets want to lend support to top-flight starting pitchers such as Jacob deGrom and Noah Syndergaard, they need a quality bullpen, and Miller could be a key piece if he manages to get over the injuries that plagued him last season.