Trump and top administration officials have expressed a desire to try to improve the U.S.-Russia relationship, which became increasingly bitter under the Obama administration.
The bill would block the administration's ability to return the compounds without congressional approval.
In addition to slapping new financial penalties on Moscow, it would give Congress 30 days — or 60 days around the August recess — to review and potentially block Trump from lifting or relaxing Russia sanctions; as well as codify the sanctions on Russia imposed by executive order by the Obama administration, and allow the Trump administration to impose new sanctions.
It would also impose new sanctions in response to Iran’s basic missile program, and target North Korea’s shipping industry and people who use slave labor.
Thursday's vote comes after Corker announced on Wednesday night that they had gotten a deal with House Majority Leader Kevin McCarthy (R-Calif.) after the House GOP push to include North Korea sanctions threatened to slow down the bill in the Senate.
Corker said that in addition to the Senate passing the Iran-North Korea-Russia sanctions bill, the House had agreed to move additional penalties against Pyongyang quickly.
Several senators had been working on their own North Korea bills, and Corker suggested this week that senators were also interested in applying congressional review to sanctions against the isolated country.
The Russia legislation has been beset for weeks over a string of setbacks.
The bill ran into a hurdle almost immediately after passing the Senate in a 98-2 vote, when the House parliamentarian said the legislation violated a constitutional requirement that all revenue bills start in the House.
After the Senate approved changes to address the constitutional issue, House Democrats then objected to a provision requested by GOP leaders that prevented them from forcing votes to block Trump from lifting sanctions.
A compromise reached over the weekend ensures that any House member can force a vote on a resolution of disapproval to block sanctions relief that has already passed in the Senate.
Updated: 6:52 p.m.