A procedural step late Monday would set up a final approval vote in the chamber in the coming days.
President Donald Trump's administration is encouraging lawmakers to back the small Balkan nation's bid.
"It is strongly in the interests of the United States that Montenegro's membership in NATO be ratified," Secretary of State Rex Tillerson told US Senate leaders in a March 7 letter.
To date, 25 of NATO's 28 members have ratified Montenegro's accession, a country of 620,000 people seen as a geostrategic ally.
The North Atlantic Treaty Organization holds its summit on May 25 in Brussels, where Trump will use the opportunity to reaffirm Washington's strong commitment to the alliance, according to the White House.
"Montenegro's participation in the May NATO summit as a full member, not an observer, will send a strong signal of transatlantic unity and that no third parties have veto power over NATO decisions," Tillerson said in his letter.
The Kremlin is opposed to Montenegro's accession, calling it a "provocation" that would reinforce the pro-Western military alliance's presence in the Balkans.
The US vote comes days after a Montenegrin special prosecutor accused "Russian state bodies" of involvement in an alleged coup plot during Montenegro's election last October.
Moscow branded the accusation "absurd."
Russia also stands accused of interfering in the US presidential election last year, when US intelligence agencies say it leaked hacked emails that damaged Democrat Hillary Clinton's campaign.
As with all international treaties, a two-thirds majority is required for final Senate approval. Success is likely, with all Democrats and most Republicans in favor of expanding NATO to include Montenegro.
Republican Senator Rand Paul strongly opposes the move, however, warning Washington against spreading itself too thinly when its military is involved in Iraq, Syria, Libya and Yemen.
"The United States is pledged to defend 28 countries in NATO," he said in a statement to AFP. "It is unwise to expand the monetary and military obligations of the United States given the burden of our $20 trillion debt."
The head of NATO's Allied Command Operations, US General Curtis Scaparrotti, told a Senate panel last week that ratifying Montenegro as a NATO member is "absolutely critical."
Besides the United States, the Netherlands and Spain have yet to ratify Montenegro's membership.
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