Montero was promoted from Triple-A Las Vegas to pitch Tuesday night against Washington.
The 26-year-old deGrom has excelled in his first season in the majors, going 6-5 with a 2.87 ERA in 16 starts. His move to the DL was made retroactive to last Friday.
The 23-year-old Montero made his big league debut earlier this year. He went 0-2 with a 5.40 ERA in four starts for the Mets. He was recently picked as the Pacific Coast League pitcher of the week after two scoreless starts at Las Vegas.