As President Obama made the case Wednesday for moving swiftly to approve a deficit-reduction deal that includes tax increases, lawmakers in both parties plotted strategies that could make it difficult to find common ground.

Senate Republicans began a push to amend the Constitution to require a balanced budget, insisting that the government cannot get out from under crushing debt without one.

In a series of speeches on the Senate floor and a morning news conference, Republicans injected the demand into talks with the White House.

Senate Minority Leader Mitch McConnell (Ky.) characterized the amendment as a “good first step” toward achieving long-term budget stability. “We think it’s pretty clear, regardless of what we’re able to negotiate here in the short term, that we should put the federal government in this kind of fiscal straightjacket so that we do not get in this position again,” he said.

Polls show that a balanced-budget amendment is broadly popular. It is also an article of faith for much of the conservative base, and the Republicans’ emphasis on its passage may be designed to show activists that they take the issue seriously.

The measure has no chance of finding Democratic support, and it would require spending cuts beyond anything under serious consideration.

In their own news conference, Senate Democrats said Republicans must accept tax increases as part of a deficit-reduction deal, which also would include raising the federal limit on government borrowing. The deadline for doing that without risking a default is Aug. 2.

Democrats called into question a statement by House Speaker John A. Boehner (Ohio) that any debt-limit deal that includes tax increases would not pass the House. Boehner has acknowledged that Congress must raise the debt ceiling even though many of his fellow Republicans won’t vote to do so.

“He doesn’t have the votes if he doesn’t have revenues, because he’s not going to get Democratic votes without revenues, and he doesn’t have enough Republican votes on his own to pass it,” said Sen. Charles E. Schumer (N.Y.), the chamber’s No. 3 Democrat.

Underlying the debt-ceiling negotiations is an unusual congressional calendar this summer. The GOP House majority opted for a schedule that has two weeks of legislative activity and then one week back home in congressional districts.

This week, the Senate is in session while the House is out, and next week the situation is reversed. Although congressional leaders will do much of the work on a deficit deal, they will need time to test the package’s provisions with their rank-and-file lawmakers, and that could be hampered by the off-and-on schedule over the next month.

Staff writer Paul Kane contributed to this report.