WASHINGTON, D.C. — President Obama told lawmakers last Thursday that he would push for a sweeping overhaul of the immigration system by early next year. But as he hosted a group of House and Senate members at a White House meeting on the issue, a new political obstacle to a deal came into view: how to regulate the future influx of foreign workers. The issue was raised by Sen. John McCain, R-Ariz., a principal architect of past attempts to rewrite immigration laws. McCain challenged Obama and other Democrats to stand up to labor unions that are pushing a plan that business groups fear could be overly restrictive in admitting future immigrant workers. “I would expect the president of the United States to put his influence on the unions in order to change their position,” McCain said following the hourlong session that included Obama, Vice President Joe Biden, senior White House officials and about 30 lawmakers from both parties. McCain’s comments came even as the White House took pains to foster the impression that the senator would be a partner in striking a deal. Obama sat directly beside McCain, his former campaign rival, during the meeting, and Obama praised the senator in his public remarks, saying that McCain had “already paid a significant political cost for doing the right thing.” Still, Obama offered no commitments on how to handle future immigrant workers, and White House officials said the meeting was not meant to be a forum for policy details. Obama did offer his firmest pledge yet as president to push aggressively for new legislation by the end of 2009 or early 2010, according to meeting participants. The president has been reluctant to offer a timeline. As his administration in recent weeks focused its attention on health care and energy, some Latino leaders and immigrant advocates cautioned that delaying on immigration could anger Latino voters who turned out strongly for Obama in last year’s election. “What I’m encouraged by is that after all the overheated rhetoric and the occasional demagoguery on all sides around this issue, we’ve got a responsible set of leaders sitting around the table who want to actively get something done and not put it off until a year, two years, three years, five years from now, but to start working on this thing right now,” Obama said. Obama said his administration was “fully behind” an immigration overhaul and that that Homeland Security Secretary Janet Napolitano would spearhead the effort. One major sticking point is whether the House would pass one of the key provisions demanded by advocates for immigrants — a pathway to citizenship for the 12 million illegal immigrants currently living in the country. Some 40 House Democrats represent conservative swing districts where there is little support for the idea. White House chief of staff Rahm Emanuel told reporters Thursday that “the votes aren’t there” to pass a plan. But it was clear Thursday that regulating the future flow of foreign workers was emerging as a new partisan flashpoint. Past plans included a temporary guest worker program that was supported both by business groups and immigrant advocates. But many labor unions were wary of that plan. Some union members fear that guest workers drive down wages and displace American workers. This year, immigrant advocates and unions pulled together to propose that an independent commission study labor market needs and decide how many immigrant workers should be allowed into the country. The commission plan has drawn opposition from business groups such as the U.S. Chamber of Commerce, and McCain on Thursday left no room for compromise in opposing it. Ana Avendano, the AFL-CIO’s point person on the issue, said the unions do not intend to give up. “Just because McCain said no (on Thursday) doesn’t mean we’re not going to continue pushing policies that are good for working people in the United States,” she said.