Rep. Mike Simpson (R-Idaho), who sits on the House Appropriations Committee, told the Hill on Thursday that subcommittee markups are set to begin next week.
But in a discussion with Minority Whip Steny Hoyer (D-Md.), Brady offered no clues as to whether or when a House-Senate budget conference might take place. Hoyer pressed Brady for an answer on when a conference might happen, but Brady said only that discussions between the House and Senate Budget Committee chairs is ongoing.
"We certainly don't want to short-circuit those discussions because we're all encouraged."
When pressed by Hoyer, Brady said only that he is encouraged because the Senate has finally passed a budget after four years without a budget.
"I'm encouraged that both sides are discussing them, trying to find a way to narrow them, and we ought to give them time to be able to continue those discussion," he said.
Democrats in the House and Senate have pushed for a conference for two weeks now, but Ryan has indicated the House will not go forward to a formal conference until the basic outline of a budget agreement is in sight. That may never happen, as the House budget includes deeper cuts with no tax hikes, while the Senate budget assumes the end of the sequester and includes higher taxes.
Hoyer also asked if the House GOP leadership had any plans to take up a bill to replace the sequester, but Brady indicated there are no plans.
"What the sequester did was take, in effect, a 500 pound government and insisted that it lose 10 pounds," he said. "That's what the sequester does."
Brady said Congress passed legislation locking in place current spending levels for 2013 and giving the administration some flexibility to get around the automatic cuts. He added that the 2014 appropriations process is the best way to address federal spending levels now.
"I think that adds extra importance to that process," Brady said.
On the debt ceiling, Brady said Republicans have started the process of discussing how best to get closer to a balanced budget. The GOP is expected to again seek spending cuts as a condition of increasing the debt ceiling later this year, and Brady said the GOP would not be looking to tax increases as part of the answer.
He also said the GOP wants to ensure a deal is in place before the last minute.
"We've begun the process of identifying good, positive ideas that would restore confidence in America's financial future, and we think it is important for us to move along in a very deliberate, timely manner so that we don't end up with an eleventh-hour issue," Brady said.