WASHINGTON (CNN) -- On the eve of a showdown over what could be a historic overhaul of U.S. immigration law, congressmen drew lines in the sand Sunday, leaving it all but impossible to envision what kind of legislation might ultimately win passage.

"It is incumbent upon us in the Senate to compromise," Majority Leader Bill Frist said on CNN's "Late Edition." He called the nation's immigration system "flat-out broken."

Sen. Lindsey Graham, a South Carolina Republican, cast the fight ahead -- and the impact it might have on Latino citizens, the nation's fastest-growing voting bloc -- as "a defining moment for the Republican Party."

The fierce debates -- centering mostly on what to do about the millions of undocumented immigrants in the country -- don't fall along a simple partisan divide. Splits within each party and a wedge between many Republicans and President Bush lay the groundwork for a passionate, bitter battle.

The Senate began deliberations Thursday and plans to try to tackle the issue this week.

Even if it manages to pass a bill this week, as Frist predicted Sunday, another fight likely would follow in the joint House-Senate Conference Committee, with some House members fighting for a tougher bill than the Senate is set to consider.

"There's a chasm between the House and the Senate," Sen. Dick Durbin, an Illinois Democrat, said on CBS's "Face the Nation."

"It's the toughest thing I've done in 37 years in public office," Rep. James Sensenbrenner, a Wisconsin Republican, told CBS. He is one of many House members opposing legislation passed last week by the Senate Judiciary Committee.

That committee's bill lays out steps for millions of illegal immigrants in the United States to obtain permanent residence or citizenship. It also strips Sensenbrenner's provision in the House bill that would make undocumented immigrants and those who help them felons.