Forget about the plausibility of the legal theory — Republicans may have a political problem on their hands by filing a lawsuit against President Barack Obama, according to a new CBS poll.

Even with Obama's approval ratings sagging, a significant majority of Americans oppose the Republican-controlled House of Representatives' plan to file a lawsuit against Obama.

In a party-line vote last week, Republicans passed a resolution authorizing House Speaker John Boehner to file the lawsuit, which will argue Obama overstepped his constitutional authority by twice unilaterally delaying implementation of the so-called employer mandate of the Affordable Care Act.

But according to the poll, 54% of Americans disapprove of the coming House lawsuit, while only 37% approve. Overall, the numbers display the fine line for Republicans ahead of the midterm elections. Two-thirds of Republicans, the base voters the party is trying to turn out in droves this November, approve of the lawsuit. But by a 53-36 split, self-described Independents oppose it.

During a news conference on Wednesday, Obama defended his use of executive actions, saying he had to act to work around congressional gridlock.

"We're going to make sure that every time we take one of these steps that we are working within the confines of my executive power," Obama said. "But I promise you the American people don’t want me just standing around twiddling my thumbs and waiting for Congress to get something done.

"Even as we take these executive actions, I’m going to continue to reach out to Democrats and Republicans — to the Speaker, to the leadership on both sides and in both chambers — to try to come up with formulas where we can make progress, even if it’s incremental."

The views on the lawsuit split with the public's overall perception of Obama, which has taken a sharply negative turn over the past year. Only 41% approve of his overall job performance. And on specific issues, the numbers are worse. Just 40% approve of how he's handling the economy, and 36% approve of his performance on foreign policy.