Democrats are pushing for another round of stimulus checks and $600 in weekly unemployment benefits

House Democrats are making another push to provide a fresh round of federally-funded unemployment benefits to replace the $600-a-week benefit that expired in July.

On Monday, Democrats unveiled a scaled-down version of the HEROES Act, their proposed stimulus package that passed in the House in May. Originally the package had a $3 trillion price tag. The revised version would cost $2.2 trillion.

It calls for extending the $600 weekly benefit through Jan. 2021 and a second round of $1,200 direct payments to households, both of which were included in the...