An amendment offered by Sen. Mike Lee (R-Utah) that would have “expressed the sense of the Senate” in support of a balanced budget amendment fell just short of passage Tuesday, but it garnered the backing of almost a dozen members of the Senate Democratic caucus, many of whom are up for reelection in 2012.

Fifty-eight senators voted “yes” while 40 voted “no” on the measure, which Lee had proposed as an amendment to a patent-reform bill making its way through the Senate.

Among the 58 “yes” votes were 10 Democrats: Mark Begich (Alaska), Michael Bennet (Colo.), Sherrod Brown (Ohio), Tom Carper (Del.), Herb Kohl (Wis.), Joe Manchin (W.Va.), Claire McCaskill (Mo.), Bill Nelson (Fla.), Ben Nelson (Neb.) and Mark Udall (Colo.).

Of them, seven are up for reelection in 2012: Brown, Carper, Kohl, Manchin, McCaskill and both Nelsons. Connecticut Independent Sen. Joe Lieberman, who typically caucuses with Democrats and is retiring in 2012, also backed the measure. All Republicans voted in favor of the measure; two Democrats, Sens. Kent Conrad (N.D.) and Mary Landrieu (La.), did not vote.

The vote is a sign that Democrats, many of whom will face tough reelection battles in 2012, are sensitive to voters’ concerns over federal spending.

Passage of a balanced budget amendment would be no easy task: It would require a two-thirds majority in both chambers and ratification by three-quarters of the states. In 1997, Sen. Orrin Hatch (R-Utah) spearheaded a move for a balanced budget amendment, winning the backing of 66 senators — including 11 Democrats — but ultimately fell one vote short of the two-thirds necessary for passage.

Republicans in the House and Senate have renewed their push for a balanced budget amendment in recent months; both Lee and Hatch have introduced balanced budget proposals, and a House measure sponsored by Rep. Bob Goodlatte (R-Va.) has attracted more than 200 co-sponsors, including a half-dozen moderate Democrats.

— Staff writer Paul Kane contributed to this report.