House and Senate lawmakers today will take their first steps toward changing the way Senate vacancies are filled, hoping to avert future controversial appointments like that of Sen. Roland Burris (D-Ill.).

This morning’s rare joint hearing of the House and Senate Judiciary committees, in which Members will hear testimony on the merits of amending the Constitution to change the Senate appointment process, could include Sen. Ted Kaufman (D-Del.), appointed to fill the seat of Vice President Joseph Biden, and Senate Majority Whip Dick Durbin (D-Ill.), embroiled earlier this year in the Burris saga, among its attendees.

While Kaufman said he thinks “it’s a great idea to have Senators elected,— he does not support amending the Constitution to change how Senate vacancies are filled.

Sen. Russ Feingold (D-Wis.), who will preside over the hearing, introduced a measure in January following Burris’ controversial appointment by now-impeached Gov. Rod Blagojevich (D). Blagojevich is facing charges for trying to sell the open seat formerly held by President Barack Obama in exchange for personal favors, and Burris acknowledged through a voluntary affidavit that he had conversations with as many as five associates of the governor prior to his appointment.

Feingold’s bill would amend the Constitution to require that special elections be held to fill any Senate vacancy. Durbin, who has urged Burris to resign, became a co-sponsor of the bill earlier this week. Sens. Mark Begich (D-Alaska) and John McCain (R-Ariz.) are also co-sponsors.

Rep. David Dreier (R-Calif.), who will offer testimony during the joint hearing today, introduced a companion bill in the House at the same time.

Despite early support of the measure, amending the Constitution is an arduous task that would require a two-thirds approval by the House and Senate. Additionally, at least 38 states would have to ratify the measure.

Rep. Aaron Schock (R-Ill.), who will also testify at today’s hearing, has a separate bill that would require special elections without having to amend the Constitution.

No matter how it’s done, the freshman lawmaker said the “debacle— that came out of his home state — in addition to the three other Senate appointments made in 2009 alone — is enough cause for legislative action.

“The point is there’s a problem with one man or one woman being in charge of who should represent the state,— Schock said.