Treasury Secretary Jack Lew told Congress the government will reach its borrowing limit around Nov. 3. (AP Photo/Ted S. Warren)

Speaker John Boehner is facing growing pressure to start work this week on a bill to increase the federal borrowing limit as the debt ceiling deadline fast approaches amid the continuing uncertainty over who will succeed the Ohio Republican at the end of the month.

The Treasury Department estimates that the debt limit will need to be raised by Nov. 3 in order to avoid a default that could rock financial markets and the economy. In order to get a bill through Congress before that deadline, congressional aides and lobbyists said Monday that they expect the House will pass an increase in the debt limit by the end of the week, with the help of Democratic votes, so it can then be cleared by the Senate next week before the deadline.

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A spokeswoman for Boehner would not comment on plans for moving debt limit legislation and cautioned that the schedule for the week remains fluid as members return to Washington after a week-long Columbus Day recess.

House Republicans will meet Tuesday night to discuss the agenda for the next two weeks and the biggest question facing the conference is whether House Ways and Means Chairman Paul Ryan (R-Wisc.) will run to replace Boehner as speaker. Ryan is expected to win the gavel if he runs, but if he passes there will be wide open race that will add further uncertainty to the legislative agenda.

The chaos surrounding who will succeed Boehner, who announced last month he was leaving Congress, has left Republicans with no clear plan for increasing the debt limit even though the once routine issue has become a critical fiscal battle in recent years. Conservatives have repeatedly threatened to block the must-pass legislation in order to force deep spending cuts.

But with the deadline looming, there is now intense political pressure from Democrats and many Republicans to increase the borrowing limit quickly to quiet fears on Wall Street and abroad that the United States could default. Lew said Monday that any failure to do so would be “ridiculous.” Senate Republicans would like to pass a big enough increase so that the issue will not have to be addressed again until sometime in 2017 when there will be a new president.

Senate Majority Leader Mitch McConnell (R-Ky), has repeatedly vowed there will be no government default and Boehner has said he would like to finish work on as many thorny political issues, like the debt limit, as possible before he steps down on Oct. 29, although he said he will stay on longer if his replacement is not in place.

Republicans in both the House and Senate have pushed for changes to mandatory spending programs — such as Social Security, Medicare and Medicaid — in exchange for a long-term debt limit increase. But time for those negotiations is running short and Democrats, including President Barack Obama, have made clear they will only support a clean increase.

White House Press Secretary Josh Earnest told reporters on Monday that spending cuts are unrelated to the debt limit, which only allows borrowing to cover the cost of expenditures already approved by Congress. The White House has frequently said that the debt limit debate should not be used as leverage for budget negotiations because of the damage a default could do to the economy.

“There’s no reason for us to engage in that kind of irresponsible brinksmanship, and it’s why Republican leaders in Congress need to accept the responsibility that they have to act without drama and delay to raise the debt limit,” Earnest said.

The House is scheduled to vote Wednesday on a contingency plan that would allow the government to “prioritize” payments if the debt ceiling isn’t increased, with interest on government debt and Social Security benefits at the top of the list.

Supporters of the bill argue this would prevent a technical default on debt while broader budget negotiations are held.

But the bill has been panned by economists and officials from both parties who argue that the government cannot pick and choose which bills to pay. In other words, a default is a default.

“Once you no longer consider all of your obligations rock solid, you’re no longer the full faith and credit of the United States,” Lew said Monday on CNBC. “It’s also not possible to pick and choose. We have about 80 million transactions a month. Our system wasn’t set up not to pay,”