The move comes just days after Kamara was suspended for an on-field spat with team-mate Frederico Higuain.
The 31-year-old had signed a contract extension with Columbus 10 weeks ago.
"It's a shock to me and my family. I have done nothing but good on and off the field for Columbus. But my journey continues and this time is for New England Revolution," Kamara said.
Columbus coach Gregg Berhalter said: "Our actions will always reflect the values of our club and we believe this transaction will improve our competitiveness this year and in future years."