As House Republican leaders prepared to meet with President Obama Thursday, conservative lawmakers were exploring the possibility of a short-term increase in the debt ceiling, perhaps trying to seize the opening after the president said he would consider the option.

Fox News has learned that Republicans are considering a proposal to raise the debt ceiling for another six weeks. They could move on it as early as Thursday, though it would not directly address the partial government shutdown.

Members of the Republican Study Committee, the most conservative bloc in the House, first said Wednesday they were looking at the possibility. Their inclination is to consider a short-term increase only if there is an agreement on a broader spending framework.

But the option could help buy time for lawmakers to nail down the specifics of a longer-term deal. The U.S. government is facing what the Treasury Department says is an Oct. 17 deadline to raise the nation's debt ceiling.

Though a short-term deal would by definition be only a stopgap fix, the development Wednesday pointed to at least a sliver of possible common ground -- something to potentially work toward, after nine days of a partial government shutdown during which lawmakers seemed to mostly talk past one another.

"Clearly, Republicans want to avoid default," Rep. Kevin Brady, R-Texas, said, adding they also want to cut spending.

Obama, speaking to reporters for over an hour on Tuesday, reiterated that he does not plan to negotiate with Republicans until a spending bill is passed and the debt ceiling is raised.

But he said he's "absolutely" willing to accept a short-term measure to fund the government and raise the debt ceiling, and negotiate with Republicans after that.

Republicans appear to still want some negotiations to take place before such a bill is passed.

The issue will no doubt be part of the discussion as Democrats and Republicans meet, separately, at the White House this week.

House Democrats met with Obama on Wednesday afternoon.

A spokesman for House Speaker John Boehner said Republicans will attend a meeting at the White House on Thursday. Though Obama invited the full caucus, a Boehner spokesman said they would only send a small group of negotiators.

"Nine days into a government shutdown and a week away from breaching the debt ceiling, a meeting is only worthwhile if it is focused on finding a solution," Boehner's office said. "It is our hope that this will be a constructive meeting and that the president finally recognizes Americans expect their leaders to be able to sit down and resolve their differences."

A meeting with Senate Republicans at the White House is set for Friday morning.

Pressure on Washington to resolve their impasse built on Wednesday as Veterans Affairs Secretary Eric Shinseki warned that millions of veterans could soon lose their benefit payments without a resolution.

Officials were also outraged after finding out that death benefits to the families of fallen soldiers had been suspended. The Pentagon, though, plans to resume payments with the help of a charity.

Fox News' Chad Pergram contributed to this report.