The White House is stepping up pressure on Congress to approve emergency measures that would reinstate payments to more than 1.3 million long-term unemployed Americans who saw their benefits cut three days after Christmas.

President Barack Obama's chief internal economics adviser appeared on two Sunday talkshows to warn against failing to reintroduce payments for those who have been out of work for more than six months – while also indicating that the benefits programme could legitimately end when unemployment rates return to “normal”.

Gene Sperling, assistant to the president for economic policy, reinforced remarks made by Obama in his Saturday address when he called the cut in benefits “cruel”.

The push by the White House comes as part of a concerted effort by senior Democrats to break the deadlock on Capitol Hill. The unemployment cheques which were cut have been paid out since the economy collapsed in 2008.

“In the last half-century we have never cut off emergency unemployment benefits when long-term unemployment has been this high,” Sperling said, on CNN's State of the Union. “We have to be a country that’s committed to leaving no one behind.”

The Senate will start discussing a fast-track bill to reinstate benefits – at least temporarily, until the end of March – as a top priority when it resumes business on Monday after the winter holiday break. A vote could be delayed until Tuesday or Wednesday, while the Senate holds a confirmation hearing for Janet Yellen, the new chair of the Federal Reserve. But the topic of unemployment is now front and centre in Congress and the debate and vote is part of a Democratic drive to bring the issue of economic inequality to the top of the political agenda this year.

The income gap is expected to be a key theme in Obama's state of the union address later this month, along with a campaign to raise the minimum wage – a subject guaranteed to ignite further opposition among Republicans who are against extending jobless benefits. Other major issues in front of legislators this year will include immigration reform and the debt ceiling.

With mid-term elections looming in November, Senate Republicans are conscious that winning just six more seats would turn over the majority to them, raising the prospect of control of both houses of Congress.

In November, US unemployment nationally hit a five-year low of 7%. Unemployment insurance for the long-term jobless expired after Congress agreed a bipartisan budget deal just before breaking for the holidays but failed to include a provision authorising the continuation of the benefits. The bill to restore them has been co-authored by a Rhode Island Democratic Senator, Jack Reed, and the Nevada Republican Senator Dean Heller, both of whose states lead the unemployment rankings at 9%.

Sperling sought to reassure Republicans – who have been reluctant to step across the aisle with Heller – that unemployment insurance is not designed as a permanent safety net to dissuade the unemployed from job-hunting. Insisting that the hardcore unemployed could only receive the benefits in question if they are actively seeking work, he said: “Our economy still has three people looking for every job [that’s available]. Most of these 1.3 million people are desperately looking.”

He gave an optimistic outlook on the economy and said that once a certain level of recovery had been achieved the emergency payments should end.

“We will start getting to a normal level of unemployment where we can cut this off. When it’s approaching 6% nationally,” he said.

Sperling is offering congressional Republicans “a grand bargain” on jobs. He has said the administration would be willing to implement corporate tax reform that lowered rates to 28% and simplified taxes for small businesses. On Sunday, he told NBC's Meet the Press that he would “do it together with a major infrastructure investment”.

Guardian data analysis, meanwhile, has shown that some of the most senior conservative Congressional Republicans, such as the Senate minority leader, Mitch McConnell, the Florida senator Marco Rubio and Senator Rand Paul of Kentucky, represent states that are among the most affected by the cancellation of long-term unemployment benefits.

Most of the 25 worst-affected states are represented by Democratic senators. However, in addition to the 34 Democratic senators feeling the most pain as a result of of the benefits cuts, there are 16 Republicans. They include high-profile figures such as Rubio, whose state has so far seen 73,000 job-seekers lose on average $231 a week.

Four in every 1,000 people lost benefits in Kentucky, which has an 8.2% unemployment rate and is represented by McConnell and Paul, who like Rubio is considered a potential presidential candidate for 2016.

On Sunday, Paul hinted that he could support the bill to restore unemployment payments, if additions were made specifically creating new jobs.

“I’m not opposed to unemployment insurance, I’m opposed to it without paying for it,” he told CBS’s Face the Nation.

Congress calculates that a year of benefits for the long-term unemployed will cost $26bn. Some Republicans have indicated that they will vote to reinstate benefits if spending cuts are made elsewhere.

The Senate majority leader, the Nevada Democrat Harry Reid, hit out at congressional Republicans on Sunday, saying their staunch opposition to extending unemployment insurance was at odds with mainstream Republican voters.

“The vast majority of the American people believe that unemployment benefits should be extended. I’m hopeful that we can get four more Republicans [in addition to Heller] to support this – gee whizz, we have never stopped benefits in this way.”