CHICAGO — In a back room at the Franciscan House of Mary and Joseph, one of the largest homeless shelters in Chicago, a social worker named Sheena Ward guided Terry Cannon through a Medicaid application.

A wet cough punctuated Mr. Cannon’s often wry answers to Ms. Ward’s questions about his disability status, military service and marital history. “I have glaucoma, I’m going blind. I have lung disease, I’m dying,” he said. “How can they deny me? If they do, give me a couple years and I’ll be gone.”

Today, most state Medicaid programs cover only disabled adults or those with dependents, so Mr. Cannon and millions of other deeply impoverished Americans are left without access to the program. But starting Jan. 1, President Obama’s health care law will expand Medicaid coverage to adults with incomes under 138 percent of the federal poverty line, and enrollment is expected to increase by about nine million next year. Thousands of homeless people will be among the newly covered.

Housing advocates say they believe that the Medicaid expansion has the potential to reduce rates of homelessness significantly, both by preventing low-income Americans from becoming homeless as a result of illness or medical debt and by helping homeless people become eligible for and remain in housing.