Harry Reid is set to meet privately with President Obama today. Many on the left expect them to discuss the Senate Majority Leader’s recent declaration of strong opposition to “fast track” trade promotion authority that will make it easier for the President to negotiate free trade deals currently being discussed with the European Union and the Trans-Pacific Partnership.

“I’m against fast track,” Reid said, adding: “Everyone would be well advised just to not push this right now.” This is being widely seen as a sign the administration will have a tough time winning Congressional approval for the fast track bill, particularly from fellow Dems — and even that it may fail.

But liberal groups who oppose fast track authority — arguing, as some economists and labor officials have, that the trade deals will mean further job loss due to globalization and worsening inequality —  are not sanguine. They argue we should be watching the House of Representative, not just the Senate, for clues to the fate awaiting fast track. The bill is supported by many Republicans and business groups, which means it would likely pass the House.

And so, liberal groups want to see a clearer sign from Nancy Pelosi and House Dem leaders that they oppose fast track authority, so that anything coming out of the House does not have a lot of bipartisan support (which would increase pressure on the Senate to act).

I asked Nancy Pelosi’s office for her stance on fast track. It turns out she addressed the issue at a recent blogger roundtable, telling those assembled that she had substantive problems with the fast track bill, and that she thought “plenty” of Democrats would oppose it. However, she stopped short of flatly declaring definite opposition, suggesting Dems were still trying to find unity behind a “kind of Fast Track Authority that we can support.”

This means Pelosi could still use her influence to shape what the final fast track arrangement looks like. But in the short term her stance won’t satisfy groups like Democracy for America and CREDO Action, which are lobbying Pelosi to declare her opposition in unequivocal terms, as Reid has, in order to send a message to other House Democrats that they should stand united in opposition. DFA tells me well over 120,000 people have now signed their petition calling on Pelosi to “send a clear and unambiguous message to her caucus that a vote for Fast Track is a vote against the Democratic leadership,” adding that fast track will make the Trans-Pacific Partnership — or “NAFTA on steroids” — possible.

This battle has created a somewhat unusual situation, in which Harry Reid is getting more accolades from progressive groups than is the reliably liberal Nancy Pelosi.

“Reid’s leadership was absolutely critical, not just for giving credibility to the anti-fast track position progressives have been pushing, but for potentially saving Senate Democrats from taking a position on job-killing trade agreements that’s incredibly unpopular with voters,” Neil Sroka, a spokesman for DFA, tells me. “Reid’s shown a willingness to stick his neck out for popular progressive positions and lead in a way the grassroots base wouldn’t normally expect to see from congressional leadership.”

However, Pelosi has a history of undercutting such fast track authority, having played a critical role in blocking it for President Bush in 2008. While that was obviously against a Republican president — and the president currently pushing for this authority is a Democrat — one assumes we haven’t heard the last from her on this issue. After all, this comes at exactly the moment when Dems are trying to make the need to combat inequality and the trends that have eroded middle class jobs and security central to their party-wide message. Keep an eye on this one.