Congressional Republicans remain skeptical of President Barack Obama’s charm offensive — very, very, very skeptical.

In conversations with House and Senate Republicans late Tuesday, a deep suspicion of his motives for reaching out and his commitment to working with them on fiscal and other issues hung like a dark cloud over their otherwise predictable comments expressing cautious optimism that Washington could be on the cusp of a new era of bipartisanship.

“There’s some concern as to whether it will be more than political window dressing,” Rep. Tom Cole, R-Okla., said during a brief interview. “I think members are happy that it’s happening. They’re not yet convinced that it’s going to be consistent and long-lasting.”

“I hope it’s genuine,” Rep. Tom Latham, R-Iowa, added. “Who knows?”

Obama’s visit comes on the same day that the president is scheduled to headline a dinner to support his political organization, Organizing for Action — only adding to House Republicans’ mistrust. This concern was raised by Chief Deputy Majority Whip Peter Roskam in Tuesday’s House GOP whip meeting, according to an individual in attendance. It was then suggested that the Illinois Republican, who served with Obama in the state legislature, should open the question-and-answer session with a query on the lack of trust.

Regardless of whether it’s the first question, sources say the trust issue is likely to be addressed.

This week, Obama will also meet with Senate Republicans. The “charm offensive,” as it is popularly known around town, began over the past few weeks with phone calls to rank-and-file Republicans who have staked out leadership positions on key issues such as immigration. It included Obama hosting a dinner at a posh downtown restaurant for a dozen GOP senators who are considered leaders on various issues — and well connected.

Republicans believe it was no accident that Sens. Richard Burr of North Carolina and Saxby Chambliss of Georgia were invited to dinner, given their close personal relationships with Speaker John A. Boehner, R-Ohio.

Some senators emerged from the gathering impressed and hopeful. But others have privately groused that Obama is less interested in establishing relationships to foster a grand bargain on fiscal issues such as the budget and the debt ceiling and is more concerned with scoring a public relations victory and ensuring the upper hand when most Republicans inevitably decline to raise income and other taxes in order to consummate a deal.

According to one GOP congressman, even if Obama is serious about seeking compromise, the gulf between House Republicans and the White House is so wide on issues such as tax and entitlement revisions, it is unclear whether a deal is possible. Where Republicans view tax and entitlement overhauls as comprehensive and multifaceted, this GOP member said, the president’s version boils down to closing a few loopholes and possibly raising the eligibility age for Medicare and Social Security.

When pressed by some Senate Republicans on various tax overhaul proposals, Obama has replied that congressional Democrats would howl in protest, according to sources familiar with last week’s dinner discussion. To underscore just how skeptical Republicans remain, even members who have been openly pleading for Obama to engage have declined to compliment the president’s initial moves as anything more than a hopeful first step.

“This is what presidents are supposed to do and have traditionally done,” Sen. Lamar Alexander, R-Tenn., said. “It’s step one in trying to solve a big problem, which is try to get to know the people that you’re working with.”

Jonathan Strong contributed to this report.