Indiana state police confirmed on Monday that the suspect, Manuel Orrego-Savala, 37, had been deported twice and was in the country illegally.
The authorities said Savala's blood-alcohol level was almost three times the legal limit when he hit Jackson, 26, and Jeffrey Monroe, 54, on Sunday morning.
Reacting to the sad occurrence, Trump threatened to go tough with the immigration policy at the border.
He wrote: "So disgraceful that a person illegally in our country killed @Colts linebacker Edwin Jackson. This is just one of many such preventable tragedies. We must get the Dems to get tough on the Border, and with illegal immigration FAST!"
In a second tweet, Trump sent his condolences to Jackson's family, writing: 'My prayers and best wishes are with the family of Edwin Jackson, a wonderful young man whose life was so senselessly taken. @Colts.'