House Ways and Means Chairman Dave Camp nudged the Obama administration to embrace a comprehensive overhaul of the tax system next year, in a speech on Thursday morning.

The comments underscored a growing GOP effort to seize the initiative on taxes and fiscal policy, after a period when Democrats appeared to dominate the political messaging wars with their focus on tax-dodging millionaires.

Republicans hope to reassure jittery voters that they’re the more responsible party on taxes and the economy, as a fiscal day of reckoning approaches at the end of the year.

Setting up a process this year for a 2013 tax rewrite “is an idea House Republicans support, the Speaker endorsed it earlier this week, and I urge the Senate and the President to get behind it as well,” Mr. Camp said at an annual tax-policy seminar. “Doing so would send a clear, strong message to the markets, to employers and families that Washington is serious about reforming our tax code and putting us on a path to sustained economic growth.”

Mr. Camp (R., Mich.), along with House Speaker John Boehner, has been seeking to build support for the idea of a 2013 tax-code revamp in recent days, in hopes of setting up an orderly timetable for addressing the nation’s mounting fiscal issues. Those issues include the expiration of Bush-era tax levels at the end of 2012; the increasing obsolescence of the basic U.S. tax code in the eyes of many experts; automatic spending cuts that will occur next year if Congress doesn’t act; and another needed increase in the federal debt limit.

The tax issues are likely the key to a comprehensive deal, but also are the most difficult to resolve, because of their sheer complexity as well as the number of affected parties – especially among businesses. That makes broad tax changes unlikely in the lame-duck congressional session, and probably necessitates at least another year of debate.

The White House didn’t immediately respond. President Obama has showed interest in overhauling the business tax system, but has been somewhat less enthusiastic about a comprehensive rewrite of the tax rules for individuals, at least on the terms Republicans are offering. The basic disagreement is over top rates. House Republicans have embraced a 25% top rate, down from the current 35%, while reducing deductions and credits.

Mr. Obama favors returning to the 39.6% top rate of the Clinton era, while adding a new minimum tax for households with incomes over $1 million and other increases for high earners.