Democrats' fortunes are "joined at the hip" with President Obama's success, a leading Senate Democrat suggested Monday.

Sen. Charles Schumer (D-N.Y.), a top lawmaker in charge of the party's messaging in the Senate, said there's a direct relationship between how Obama is faring and how Democrats will do.

"We Democrats in the House and Senate know we're joined at the hip with the president," Schumer said on "Good Morning America" on ABC. "He does well, we do well, and vice versa."

Schumer's admonition comes after weeks of bloodletting between congressional Democrats and the White House in the debate over Obama's tax-cut deal. The administration managed to successfully push its deal with Republicans through the House and Senate, but not without loud — and sometimes embarrassing — public squabbles with liberal lawmakers, who accused the administration of being too weak in negotiations.

The White House must be prepared to draw more lines in the sand when it faces down a Republican-controlled House come January, Schumer counseled.

"My advice to the president is compromise when you can, but when people are being unreasonable … draw some lines in the sand and fight," he said.

One issue on which Obama is drawing such a line is the New START Treaty, for which Schumer said the White House was pushing hard in the closing days of the lame-duck Congress.

"The president is working really hard on this. He's burning up the phone lines," he said. "We do need nine or 10 Republican votes on this, and I do think we will get them."

The Senate will go into a closed-door session Monday to debate the treaty, and a test vote to see how much support the treaty enjoys could be held on Tuesday.

New Republican lawmakers are sworn in on Jan. 5. In the interim, lawmakers are rushing to finish lame-duck action on several outstanding bills, including the New START Treaty and a bill to fund health programs for first responders to the 9/11 attacks, both of which face Senate action this week.

Schumer said he believes Democrats have the votes to advance the 9/11 bill, and said House Speaker Nancy Pelosi (D-Calif.) appeared likely to bring back the House for a final vote on that measure.