Greinke, 28, has a personal 15-game home winning streak and is 9-3 with a 3.44 ERA in 21 starts overall this season.
"There are so many good things about him, it's going to be difficult when we trade him," Melvin said.
He said the decision to trade Greinke was made after the Brewers went 0-6 on their most recent road trip. Milwaukee went into a home game against the Washington Nationals Thursday with a 44-53 record, 13 1/2 games behind first-place Cincinnati in the National League Central.
Greinke is 85-76 with an ERA of 3.79 in 259 career appearances -- 218 of them starts.
The non-waiver trading deadline is July 31.