WASHINGTON—With deficit talks kicking off in earnest, Democrats are divided on the magnitude of changes they would accept when it comes to overhauling Medicare and other safety-net programs.

The party is split between those who would agree to major adjustments, including increasing premiums for wealthier beneficiaries and raising Medicare's eligibility age, and those who rule out such moves altogether. In the middle is a group that would tolerate some cuts as long as they didn't hit beneficiaries directly.

White...