Emboldened by the response to President Obama’s face-off with House Republicans last week, the White House is intensifying its push to engage Congressional Republicans in policy negotiations as a way to share the burden of governing and put more scrutiny on Republican initiatives.

Top aides say Mr. Obama intends to follow through quickly on his State of the Union proposal for bipartisan White House brainstorming sessions. Republicans will also be invited to the White House this weekend to watch the Super Bowl, as well as to Camp David and other venues for social visits.

The outreach represents a marked shift in both strategy and substance by Mr. Obama and his allies at a time when Democrats are adapting to the loss of their 60-vote supermajority in the Senate and the president has been losing support among independent voters.

The White House’s goal is to show voters that Mr. Obama is willing to engage Republicans rather than govern in a partisan manner while forcing Republicans to make substantive compromises or be portrayed as obstructionists given their renewed power to block almost all legislation in the Senate.