House Republican leaders on Thursday made it clear that their chamber will proceed with an immigration overhaul on its own terms.

With the Senate set to consider its tenuously negotiated immigration rewrite as early as the first week of June, the top four House Republicans — Speaker John A. Boehner of Ohio, Majority Leader Eric Cantor of Virginia, Majority Whip Kevin McCarthy of California and Republican Conference Chairwoman Cathy McMorris Rodgers of Washington — issued a joint statement Thursday that was co-signed by House Judiciary Chairman Robert W. Goodlatte of Virginia.

The statement appeared to be an effort to tamp down the widely held notion in the Senate that passage of that chamber’s bill would put inordinate pressure on the House to take it up and approve it.

“While we applaud the progress made by our Senate colleagues, there are numerous ways in which the House will approach the issue differently,” they leaders said. “The House remains committed to fixing our broken immigration system, but we will not simply take up and accept the bill that is emerging in the Senate if it passes.

“Rather, through regular order, the House will work its will and produce its own legislation,” they continued, adding that each chamber should “engage in a robust debate and amendment process.”

“Our nation’s immigration processes, border security, and enforcement mechanisms remain dysfunctional [and] the House goal is enactment of legislation that actually solves these problems,” they concluded.

But while the Senate managed in a days-long markup to advance its immigration bill, the House still appears to be struggling to settle on a strategy.

A bipartisan group of House members, mirroring the Senate’s “gang of eight,” has been meeting consistently over the past several weeks to craft a broad-sweeping overhaul measure. But on Thursday, as members prepared to leave town for the recess, it seemed unlikely they would meet their self-imposed deadline to work out an agreement before Republicans strike out on their own, Idaho Republican Rep. Raúl R. Labrador suggested. Labrador has been a key negotiator in that group.

Meanwhile, some Republican lawmakers have embraced a piecemeal approach to immigration policy changes, which they say yields a better chance of success than a huge bill where a single unpopular provision could bring the whole thing crashing down.

“I think everyone, if they really think about it, supports [this approach],” said Rep. Darrell Issa, R-Calif., who, as chairman of the Oversight and Government Reform Committee and a member of the Judiciary Committee, joined Goodlatte on Thursday in introducing legislation to improve high-skilled immigration programs. “You shouldn’t have a bad item traded for another item in another part of the bill.”