The Senate approved the legislation by voice vote Thursday while the House was passing a different version on a 399-19 vote.
The votes are a show of solidarity with President Barack Obama, who has already announced sanctions against Russian President Vladimir Putin and others.
Lawmakers intend to get a final measure to the White House by day's end.
Each bill would provide $1 billion in loan guarantees to cash-strapped Ukraine and sanction Russia for its annexation of Crimea.