The measure (House Bill 1274) is expected to come up for a vote Monday, two days before lawmakers must adjourn.
The bill passed the House by a single vote, and another close vote is expected in the Senate.
Both sides are planning to use the weekend to try to shore up support, and both are mainly focused on lobbying Democrats. Republicans have largely opposed the bill.
Gov. Bill Ritter won't say if he's leaning one way or the other on the bill should it reach his desk, but he has said he's open to listening to all sides before making a decision. Ritter is a former district attorney who has requested the death penalty in prosecuting cases.
The bill would take the $1 million now being spent to prosecute death penalty cases and use it to investigate cold cases. That would add seven employees to the state's cold case unit, which currently has only one investigator.
A group of families of murder victims whose cases remain unsolved are the main force behind the bill, but they have the support of the Colorado Catholic Conference and the American Civil Liberties Union.
The Colorado District Attorneys Council is fighting the bill along with some victims' groups.