GOP Leader Mitch McConnell called it a "partisan effort to weaken the First Amendment."
Votes on border legislation, highway funding and reforming the VA are expected.
“There will be no weekends off,” the Democratic leader said.
The House will vote on a measure providing $659 million to address the surge of child migrants crossing the border.
The House will vote Wednesday on the resolution authorizing the GOP lawsuit against President Obama for his use of executive action.
The Senate starts at 10 a.m. on Tuesday and will vote to confirm Robert McDonald to be Veterans Affairs Secretary at 2:45 p.m. Later in the day it will also vote to extend the Highway Trust Fund through May
The Senate is due to vote on the confirmation of Robert McDonald at 2:45 p.m.
The Senate could vote Wednesday on the legislation.
With only a week left to pass legislation to fund the Highway Trust Fund and border security before the August recess, lawmakers will spend Monday debating nominations and suspension bills.
The House will consider a measure to overhaul the child tax credit by requiring applicants to provide their Social Security numbers. Such a change would effectively prevent undocumented immigrants from qualifying for the $1,000 refundable child tax credit.