“They don’t have anything positive that they want to talk about,” said Rep. Chris Van Hollen (D., Md.). “They will talk about [the health law] until the cows come home.”

In recent days, House GOP leaders signaled they are taking a go-slow approach on a tax-code overhaul. GOP aides said the bill could become an unwelcome distraction for Republicans, drawing public attention away from the health-law imbroglio.

In the tax-writing Ways and Means Committee, a tax overhaul already was considered a heavy lift because it requires difficult trade-offs between cutting taxes and eliminating deductions. The panel has other important issues on its plate as well, including major trade deals on the horizon.

Republicans on the committee “have maintained a complete focus on undermining the Affordable Care Act,” said Rep. Sander Levin (D., Mich.), the panel’s top Democrat, in a recent letter to Chairman Dave Camp (R., Mich.). “That obsession has come at the detriment of so many other important issues that should be the focus” of the committee, Mr. Levin wrote.