Yet, since taking the gavel of the influential Ways and Means Committee, Chairman Paul Ryan has emerged as President Obama's most convincing messenger on Capitol Hill on behalf of a free-trade agreement that would be a big boost to the president's legislative legacy.

The House could vote on Trade Promotion Authority, also known as "fast track," as early as this week. The legislation would limit Congress to an up-or-down vote on future trade deals negotiated by the president. If it passes, it will be a kind of graduation moment for Ryan, the first major legislative accomplishment since he left the Budget Committee. Unlike his visionary budget documents that made him a sensation in conservative circles, however, trade puts Ryan on the same side as a president conservatives distrust and tests the relationships he has carefully built with those in the far right of his party.

"The fact that the president is involved in the process makes it more difficult for any Republican, compared to simply putting together our own budget that the president does not have to sign," House Financial Services Committee Chairman Jeb Hensarling says of Ryan's trade push.

Since taking over the House Ways and Means Committee six months ago, Ryan has been engaged in mapping out the way forward on a trade bill that is a key priority for Obama. But Ryan's calculation, and the message he gives to Republicans, is that it is just as important to the GOP's own economic and electoral success.

"If Republicans derail trade, we add to the narrative that America is on decline, which is now uniquely tied to Obama's policies—policies we disagree with profoundly as Republicans," Ryan told National Journal.

Even before Republican leaders began whipping the bill weeks ago, Ryan was undertaking his own member-tracking operation. He held dozens of education seminars. He invited conservatives to meet with him privately. He listened to concerns and made his pitch that trade was an essential part of growing the economy. This week alone, Ryan has met with 20 members of his party in one-on-one meetings. While he acknowledges he's working on the same side as the president, Ryan underscores that the trade bill he is pushing will outlast this president, and that trade deals are fundamentally conservative.

"If the name Obama was nowhere to be seen in this, it would be fine for most people," Ryan says. "We believe in free trade. That is one of our party's primary principles."

As the vote on TPA approaches, there is increasing pressure on Ryan and Republican leadership to deliver votes from conservatives who are skeptical of supporting anything that boosts Obama. While the White House has its own whip operation, it's been difficult for administration officials to garner the 25 to 30 Democratic votes they will need to deliver. Several GOP aides have expressed frustration that the administration isn't holding up its side of the bargain. Many have blamed the administration for leaning on members it had—up to this point—failed to build relationships with. And to make things more difficult, trade unions have stepped up their lobbying efforts and Democratic leaders are staying on the sidelines.