Washington(CNN) Vice President Mike Pence's attempt to peel away House Republicans to support the White House's tax plan spurred a backlash among a critical bloc of conservative lawmakers Wednesday, Hill sources said.

Pence pressed Republican Study Committee Chairman Mark Walker to poll his members to see if they would split with House Speaker Paul Ryan and support the White House tax plan -- which lacks a border tax that is the centerpiece of the House GOP proposal.

But in a flurry of exchanges Wednesday afternoon, Republican Study Committee members turned on the White House, saying they should not blindly accept whatever President Donald Trump is selling, the sources told CNN.

The drama started with an 11:30 a.m. text Wednesday from Walker to members of the group's leadership, saying that Pence asked him to get RSC members to break with Ryan and support the White House tax plan.

A heated discussion ensued over text, with some RSC members suggesting they should support the President -- and multiple others saying it was their job to be an independent branch of government and saying that they should not abandon House leaders.

As part of the exchange, Walker said he told Pence he would have to talk with House Ways and Means Committee Chairman Kevin Brady -- a key architect of the competing House Republican tax plan -- before talking to his membership.

One source who reviewed the text exchange between Walker and the RSC members, said the request from Pence seemed out of the ordinary.

"My eyebrows raised when I saw that text," said the source.

A White House aide said that Pence met with Ryan, Brady and Walker individually at the Capitol this week. The aide said that the meetings were designed for Pence to take the pulse of House Republicans and get an idea of where they could find compromise.

"The vice president was talking to leadership, as well as other members in terms of: 'Where is your membership? Where is leadership? Where are you?'" the White House aide said.

Walker downplayed the request from Pence when asked about it Wednesday.

"I don't know if I would call it an aggressive lobbying process. I think they were concerned, as they should be, about getting some things through the House that they promised on. And I think that's one of them," Walker said.

A Ryan spokeswoman denied there was any friction between House Republicans and the White House.

"House and Senate Republicans, and the White House, are jointly working on a tax reform proposal that we can all coalesce around," said AshLee Strong, a Ryan spokeswoman.

Republicans have established tax reform as the next major piece of their agenda that they would like to get to the President's desk -- but compared to the health care battle, pushing a sweeping tax bill through Congress appears to be a Herculean lift.

Lawmakers and the White House have already split on whether a border adjustment tax -- which would levy a new tax on imports to pay for other tax cuts -- should be included. The border tax is a centerpiece of the plan Ryan has been working on for close to a decade, but the White House has shunned the idea.