When House lawmakers come back into session Thursday to finish their budget deliberations, they'll also take part in what's become a tradition of sorts on Beacon Hill: changing the law to make sure only Democrats get appointed to open U.S. Senate seats.

Legislation from Amherst Representative Mindy Domb would force Gov. Baker to name an interim appointee from the same party as the out-going senator, meaning that Sen. Elizabeth Warren would be replaced by another Democrat should she resign to join the Biden administration.

"The legislation is at its essence, is saying we're going to make sure that the party of the person is the same party as who the voters put in there in the first place," Domb told GBH News.

In the last two decades, the law has been changed by the Democrat-controlled legislature a handful of times to make sure Republican governors can't name a Republican senator, and then changed back or altered when a Democrat sits in the corner office.

"I don't deny that actually it is partisan, because I believe that the voters in Massachusetts make partisan choices," Domb said of her amendment.

Former interim U.S. Senator Mo Cowan was the beneficiary of the current law, which lets the governor name any placeholder until a special election.

"It's important to make sure that folks across the Commonwealth have the full complement [of] representation in the congressional delegation, particularly at the beginning of an administration," Cowan told GBH News.

"I think it would be helpful to have two senators in the chamber. Clearly, there's going to be a lot of work, important work, that the new administration is going to want to focus on," Cowan said.

Whether Warren is even being vetted by the Biden team is unclear, since the senator, a former presidential candiate who campaigned on Medicare for All and reigning in private equity firms, might have a difficult confirmation a Republican-controlled Senate.

Massachusetts House Democrats will weigh in on the proactive change Thursday or Friday during their annual budget debate.