But Boehner, R-Ohio, added that firing Shinseki won't solve the VA's systemic problems.
"I've not called for Gen. Shinseki to resign — although I have to admit I'm getting a little closer," Boehner told reporters Thursday. "But here's the point; this isn't about one person. This isn't about the secretary. It's about the entire system underneath him.
"I don't want people to get confused about what the shiny ball is here. The shiny ball is a systemic failure of this agency."
The VA Inspector General's office this week reported that 26 VA facilities are being investigated nationwide, including a Phoenix hospital where 40 veterans allegedly died while waiting for treatment as staff there kept a secret list of patients waiting for appointments to hide delays in care.
Despite Boehner's reluctance to call for Shinseki's head, the speaker suggested that new reports of VA misconduct makes it difficult for him to indefinitely back the retired four-star Army general.
"The reports that continue to come are appalling. And these are men and women who served our country — we've not just let them down, we've let them die," Boehner said.
"This is awful stuff. And somebody ought to be held accountable for it."