Obama ruled out ending the requirement for purchasing of health care, a part of the health law Republicans repudiate as a government overreach. But he pointedly did not reject repeal of the tax, which many Democrats as well as Republicans have already signaled they are ready to jettison.
Republicans are also expected to mount a major attack on federal deficits.
With lawmakers planning to return to Washington next week, Obama invited congressional leaders to a meeting on Friday.
Obama could use the president's veto power if Republicans pass bills he opposes, such as a repeal of the health care law. Overriding a presidential veto requires a two-thirds vote in each chamber of Congress, an unlikely scenario.
McConnell has been a severe critic of Obama, but has also helped broker bipartisan deals that ended last year's government shutdown and twice averted federal default.
He promised Wednesday that “there will be no government shutdown or default on the national debt.” Still, he said veto showdowns are also possible in the two-year era of divided government just ahead.
McConnell, 72 and famously taciturn, smiled and joked with reporters one day after the fulfillment of a lifelong dream. He and House Speaker John Boehner will have the authority to set the congressional agenda.
Republicans picked up at least seven Senate seats, giving them at least 52 in the 100-member Senate. They could gain two more in uncalled races in Alaska and Virginia, and a third in Louisiana, which headed for a Dec. 6 runoff.
Republicans had made Obama's presidency the core issue of their campaigns, even though he wasn't on the ballot. They tapped into a well of discouragement at a time many Americans are upset with a sluggish economic recovery and are besieged by troubling news, such as the spread of Ebola and the rapid rise of Islamic State militants in Iraq and Syria.
Nearly two-thirds of voters interviewed after casting ballots said the U.S. was seriously on the wrong track. Only about 30 percent said the U.S. was headed in the right direction.
In the House, Republicans were on track to meet or exceed the 246 seats they held during Democrat President Harry S. Truman's administration more than 60 years ago.
