The bill, which includes provisions from President Joe Biden's initial $2.3 trillion infrastructure proposal, authorizes additional spending for roads, bridges, highway safety, electric vehicle charging stations, rail, transit, drinking and wastewater infrastructure.
The other formalizes the legal status of migrant farmworkers -- who the White House notes are "living in the shadows and vulnerable to exploitation" -- and offers them an opportunity to earn citizenship.
"It's really just a matter of paperwork. But we are going to have a vote as soon as we can," Democratic Representative Katherine Clark, the assistant House speaker, told CNN.
The House was still awaiting the Senate version late on Monday. But House Speaker Nancy Pelosi told reporters that lawmakers would "take it up Wednesday morning at the latest."
Police use of force has been closely scrutinized following the May 2020 death of a Black man named George Floyd after a Minneapolis officer knelt on his neck.
The Cincinnati, Ohio-based 6th US Circuit Court of Appeals in 2020 granted the officers qualified immunity, ruling that no "clearly established" precedent showed that their actions were unlawful.
The other bill extends the initial background check review period to 10 days from three. Under existing law, the sale of guns can proceed if a background check is not completed within three days.
I look forward to signing it into law ... so that together we can strengthen and restore American democracy for the next election and all those to come.