After signaling earlier in the day that his leadership style could lend itself to passage of immigration overhaul without the votes of a majority of his own party, Speaker John A. Boehner, R-Ohio, took some heat from outside conservative groups.

The Conservative Action Project on Tuesday afternoon circulated a letter to House Republican lawmakers co-signed by dozens of influential movement leaders, urging them to formally adopt the “Hastert Rule.”

Named for Rep. J. Dennis Hastert, R-Ill., who served as speaker from 1999 to 2007, the rule dictates that no bill come to the floor without assurance that a majority of GOP members will vote for it.

Calling it “one way past Speakers … ensured that Republicans stayed true to their mandate,” the letter’s co-signers expressed dismay that under Boehner’s leadership, the GOP House leaders have lined up bills that “threaten to divide Republicans and in doing so empower the liberal minority of the House.”

The letter, in part, reads:

Recently House Republicans have passed bills that are inconsistent with its mandate from the American people…. The fiscal cliff tax increases, increased pork spending buried into the Hurricane Sandy relief bill, and the Violence Against Women Act reauthorization all passed the House over the opposition of a majority of House Republicans.

…

We are writing you today to encourage you to boldly use your majority not only to present a positive conservative vision, but also as the last backstop against the worst excesses of liberalism and Washington deal-making. Liberal Democrats control the White House and the Senate. We should not help their cause by handing them the keys to the House as well.

The letter also indicated that many current Republican House members support the Hastert Rule, but did not specify those members.

Boehner spokesman Michael Steel disputed the premise of the letter. “Speaker Boehner has been clear:  our goals is always to pass legislation with a strong Republican majority,” he said in an email to CQ Roll Call.

Boehner, in an interview Tuesday morning on ABC’s “Good Morning America” with news anchor George Stephanopoulos, indicated that, while he didn’t “believe that would be the case,” his approach as speaker could facilitate passage of an immigration bill — and a provision ensuring a pathway to citizenship in particular — without the majority of rank-and-file Republicans on board.

“I’ve allowed the House to work … well, more than any speaker in modern history, to the point where there are some bills that have passed with a majority of Democrats in favor, and a minority of Republicans,” he said. “And I’ve been criticized for it. What I’m committed to is a fair and open process on the floor of the House, so that all members  have an opportunity.”