The way the forced vote would happen is lawmakers in the House can offer "privileged" motions on impeachment, which means they can force the House floor to consider the subject.
By doing that, it forces Democrats to go on record about wanting to impeach Trump from office.
Green argued that Trump should be impeached for his rhetoric against certain racial and ethnic groups in the US, including Latinos, Muslims and African-Americans.
Trump's rhetoric, while not criminal, still "divides and damages the social fabric of our country in ways that obstruction of justice cannot," Green wrote in his letter.
He said that the founders wanted a way to be able to remove a President who was "causing harm to society," adding that impeachment "was designed for a time such as this and a President such as Trump."
White House spokesman Raj Shah called Green's effort "disappointing" in a statement to CNN.
"It's disappointing that extremists in Congress still refuse to accept the President's decisive victory in last year's election," he told CNN. "Their time would be better spent focusing on tax relief for American families and businesses, and working to fund our troops and veterans through the holiday season rather than threaten a government shutdown."
Top Democratic leaders have repeatedly urged Green to refrain from bringing this up on the floor, arguing this is not the time to push impeachment.
A senior House Democratic leadership aide told CNN that "Pelosi and Hoyer will vote to table," referencing House Minority Leader Nancy Pelosi and Maryland Rep. Steny Hoyer, the second ranking Democrat in the chamber.
The aide added, "They've been clear and consistent that now is not the time for this vote and we should be giving time for (special counsel Robert Mueller's investigation) to play out. "
There will be no vote directly on the resolution, so the vote to table is effectively the vote on impeachment -- some Democrats are expected to ignore leadership and oppose tabling but it's unclear how many will do so. The base wants Democrats to move on this issue so it's a tough vote for a big chunk of the caucus.
In May, Green called for the impeachment of Trump, making him the first member of Congress at the time to officially request leveling charges against the President from the House floor.