Tea Party favorite Senator Rand Paul is expected to introduce an amendment Tuesday that would require "a vote on a clean raising of the debt ceiling," the Kentucky Republican's spokeswoman, Moira Bagley, tells Fox.

The move is designed to demonstrate, as House Republicans did with a similar vote in that chamber last week, "that the votes don't exist," according to Bagley, who said support is "growing" among Republicans.

Few Democrats have expressed public support for such a vote, though leadership has not taken that option off the table. And last week, in a conference call with reporters, one senior Democrat, Sen. Tom Harkin of Iowa, voiced support for such a move. " I would prefer a clean debt ceiling vote," Harkin said.

Still, a number of nervous Democrats, along with all Republicans, have said they would refuse to increase the nation's $14.3 trillion debt limit without a deficit reduction package in place. Polls show support for a debt ceiling increase unpopular with Americans.

The Senate roundly rejected a plan recently by Paul, however, that would have balanced the federal budget in five years, in part by eliminating three Cabinet agencies. The rejection came after the Senate also voted down a House GOP budget that contained a controversial plan to remake Medicare. Democratic leaders forced that roll call vote, they said, to demonstrate to Republicans that support was not sufficient.

As August 2 fast approaches, the date by which the Treasury Department says the nation's debt ceiling needs to be raised, lawmakers are trying to find a compromise on slashing the nation's soaring debt, while votes in both chambers, designed to put members on the record on difficult provisions, continues.

It is unclear if Paul would get a vote on his amendment.