Members of the bipartisan Senate group working on a sweeping immigration bill said they were rushing to introduce a bill as early as possible, with Sen. John McCain, R-Ariz., suggesting that the measure could be wrapped up by the end of this week.

The newfound bullishness comes as senators and their aides are trying to reach consensus on some sticking points, among them a new visa program for agricultural workers, visas for high-tech workers and family unification visas.

“We’re still aiming for this week,” McCain said. Sen. Lindsey Graham, R-S.C., another member of the group, added, “If everything holds the way it is today, we’re very close.”

Sen. Jeff Flake, R-Ariz., said the group was working to resolve the question of granting new temporary visas for highly skilled workers without harming American workers. “I wouldn’t characterize it as a big disagreement, but we’re still working on it,” he said.

Flake said he and other Republican members of the bipartisan group are expected to brief the GOP Steering Committee on the details Wednesday. “We have a regular [Wednesday] policy meeting at lunch, and it will be discussed at that time,” he said. Asked whether the full details of the bill would be provided, he said, “Such as we have.” Members have not yet received the legislative language back from staff who are drafting it, so it’s unclear how thorough the briefing will be, according to a source.

Sen. Charles E. Schumer, D-N.Y., also a group member, said, “I believe we are on track” to produce a bill by the end of the week.

The Democratic members of the group spoke at their caucus luncheon Tuesday and “provided an update on the process,” according to a Senate Democratic source.

The group has been working on the bill behind closed doors since January and has been criticized by some senators for not sharing details.

Sen. Jeff Sessions, R-Ala., who has been chief among those critics, said he was disappointed that the Judiciary Committee, which has jurisdiction over the issue and of which he is a member, will not get an advance look at a bill it will have to consider before the entire Senate does. “The system doesn’t work well when a small group of senators produce a product and you have a very short period of time to review it,” he said.

His comments came after Sen. Charles E. Grassley, R-Iowa, the ranking member on Judiciary, wrote to the Republican members of the group of eight, asking that their staff be put in contact with GOP committee staff by the end of Monday. That didn’t happen.

Immigration advocates and observers have been eagerly awaiting the bill for several weeks. The Senate group has repeatedly pushed back deadlines for introducing it.

Labor unions and farm groups also have been negotiating an agreement to create new visas for agricultural workers but have yet to settle on the number of foreign workers who would be admitted under the program and what they would be paid.

Even as the group’s members remained tight-lipped, other congressmen were vocal. Senate Minority Whip John Cornyn, R-Texas, introduced a border security bill along with House Homeland Security Chairman Michael McCaul, R-Texas, and Rep. Candice S. Miller, R-Mich., on Tuesday that he said would require the Department of Homeland Security to come up with a way to gauge how secure the border is. “They currently have no metric,” Cornyn said. “They have no measuring stick, no yardstick to measure border security.”

Cornyn said he hoped his proposal would find a home in the comprehensive bill. “I hope the ‘gang of eight’ and others will be informed by that bill,” he said. “I know they have similar frustrations with the Department of Homeland Security.”

Advocates also are mobilizing for a Capitol Hill rally Wednesday, at which they will demand that the roughly 11 million undocumented people be put on a path to citizenship. They will also demand that the Senate bill make it easier for family members of U.S. citizens and legal permanent residents to live and work in the United States. Right now, backlogs force some people to wait more than two decades before securing a green card, a process that advocates say unnecessarily tears apart families.

Family reunification visas have long been a priority for Sens. Richard J. Durbin, D-Ill., and Robert Menendez, D-N.J., both of whom are part of the Senate immigration group.

But the group is likely to replace some family visas with employer-sponsored green cards, Graham said. Business groups have been clamoring for new visas for years, saying they are unable to fill high-tech jobs with only U.S. workers.

“There will be an economic-based immigration system replacing the family-based immigration system,” he said. “There will be a family component but no more chain migration.”