Yes, it is possible to expedite a green card application based on humanitarian grounds, but such cases are often complex and require substantial evidence. 

Things such as detailed Explanation of Humanitarian Grounds: Clearly articulate why your situation is humanitarian in nature. This could include evidence of severe hardship or dangerous conditions in your home country or personal circumstances that necessitate urgent processing.

Personal statements, personal affidavits or statements from you and any family members or witnesses explaining your situation in detail. pending deportation order or evidence of unsafe living conditions.