A delay in finishing work on a comprehensive immigration overhaul could threaten the recent momentum for Congress to take on the issue.

A bipartisan group of eight senators has been working since January on a bill, and they expect to unveil their package the week of April 8.

But the group has come under criticism recently from prominent Democrats and Republicans — including Judiciary Chairman Patrick J. Leahy, D-Vt., and ranking member Charles E. Grassley, R-Iowa — as well as advocates of immigration policy changes. The complaints range from the group’s decision to craft the bill behind closed doors to the failure of the eight to release their proposal in March, as they had previously pledged.

And even though members of the “gang of eight” say they are on track to release a bill in two weeks, their public pronouncements are more hopeful than definitive.

On the flip side, the group’s effort appears to be gaining ground with key conservatives. Just this week Sen. Rand Paul, R-Ky., came out in favor of comprehensive policy changes, including providing a path to citizenship, and the Republican National Committee urged support for tackling an overhaul.

Senate Majority Whip Richard J. Durbin, D-Ill., a member of the immigration group, stressed that it’s a complex issue and it has taken time to find a bipartisan solution.

“I think most of our colleagues understand the complexity of this challenge,” Durbin said. “This is not a matter that you can just put together a bill, put it in a committee and expect it to be reported out.”

Durbin added, “We are still working on the measure and we are hoping to come to an agreement on most parts of it before we leave [for recess] and when we return, if everything works well, we’ll be able to announce a bill.”

A major immigration rally on the West Front of the Capitol is planned for April 10, and the Senate group may want to present its measure before that.

Taking too long to draft a measure or get the bill to the floor could threaten its support — particularly among the Republicans needed to pass it — by giving opponents time to find flaws with the package or the process.

Durbin said members of the group hope to talk to Leahy, who said in a statement Wednesday that he believes the Judiciary Committee will not be able to take up the measure until May, because of the group of eight’s failure to unveil it in March.

“We are going to speak to Sen. Leahy and we hope we can convince him that this is a matter of urgency that he should take up as quickly as he can,” Durbin said.

One possible timeline, if the bill is unveiled April 8, could have the committee set a markup for April 11, a Senate aide said. Any senator on the committee can delay committee business for one week, and usually does. That would push the markup to April 18, setting up possible Senate passage by the Memorial Day recess.

But the timeline is far from certain. Leahy and six Republicans who recently wrote to the chairman — including Grassley — have expressed concern about dealing too quickly with the measure. They want time to read the legislation and offer ample amendments, and they don’t want to feel as if they are getting jammed.

One Democratic aide said the group is not trying to rush the committee and expects that there will be plenty of input, both from committee members during a lengthy markup and from the entire Senate where the floor debate will likely last weeks.

Asked about Leahy’s frustration with the pace Wednesday, a GOP member of the group of eight had just two words for the chairman. “Be patient,” South Carolina Sen. Lindsey Graham said.

Meanwhile, a group of about 40 pro- immigration activists led a protest Thursday outside the office of Sen. Charles E. Schumer, D-N.Y., another member of the bipartisan group of eight.

Elizabeth Alex, director of central Maryland voter engagement for Casa In Action, said the group wanted to hold the Senate group to its self-imposed March deadline.

“We don’t want this to get pushed to the back burner. Every day that they wait, more and more immigrant families are being separated,” Alex said. “So we just came today to remind them that the time is now. They need to do what they promised to do.”

Schumer, coming out of a news conference on the budget resolution Thursday, said he did not know protesters were at his office but that he is sympathetic to their concerns and their desire for a bill to get done quickly.

“I understand people’s frustration. People have waited a long time,” Schumer said. “But we are real close for the first time to coming up with a bipartisan agreement that has a darn good chance at becoming law, and we’ll need all the support we can get.”

Schumer emphasized that the group is on pace to meet its target for legislative language by the end of March.

“We met this morning for two hours. We’re meeting again this afternoon.” Schumer said. “We’re on track.”

David Harrison contributed to this report.