On Wednesday, Bob Nightengale of USA Today reported both teams are "showing strong interest" in Minor, who owns a 2.60 ERA through four starts in 2019.
The 31-year-old left-hander struggled in his first start of the season, allowing six earned runs across 4.2 innings against the Chicago Cubs. He's been dominant since, giving up just two earned runs in 23 innings (0.78 ERA) in three outings so far in April.
That includes a complete game shutout of the Los Angeles Angels on Tuesday night. He struck out seven while allowing only three hits and two walks.
"I think I have better stuff now than I did back then," Minor told reporters after going at least seven innings in three straight starts for the first time since 2014. "That's what you want every time, to go deep into game, give the bullpen a day off and help out the other guys."
The Phillies and Mets could both benefit from adding more depth to their starting rotations in what's projected to become a heated four-team race in the NL East with the Braves and Washington Nationals.
Philadelphia ranks 22nd in starter ERA (5.10) in the season's early stages, while New York is 25th (5.62) in that category, per ESPN.com.
The Rangers are expected to endure a rebuilding campaign—FiveThirtyEight forecasts them to finish with a 73-89 record—but may want to wait to see whether they can build off a solid 9-7 start before beginning to move their veteran assets.
If the selling process does begin, Minor figures to become one of the most coveted pieces, as the early interest from the Phillies and Mets showcases.